Let $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ be $CO$-open and
$\mathcal{L}$ a generalized microlocal Lagrangian with normally hyperbolic
linearized equations. Then the Peierls bracket associated to $\mathcal{L}$
extends to $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$, has the same support
property of Proposition 2.4.5 and depends only locally on $\mathcal{L}$, that
is, for all $F$, $G\in\mathcal{F}_{\mu c}(B,\mathcal{U},g)$,
$\\{F,G\\}_{\mathcal{L}}$ is unaffected by perturbations of $\mathcal{L}$
outside the right hand side of (2.59). The same locality property holds for
the retarded and advanced products.

###### Proof.

Clearly $\\{F,G\\}_{\mathcal{L}}$ is well defined, in fact since
$\mathrm{WF}(G^{(1)}_{\varphi}[0])$ is spacelike, and $\mathcal{G}_{\varphi}$,
according to Theorem 1.2.16, propagates only lightlike singularities along
lightlike geodesics, then $\mathcal{G}_{\varphi}dG_{\varphi}[0]$ must be
smooth, giving a well defined pairing. As for support properties the proof can
be carried on analogously to the proof of Proposition 2.4.5.

We now study the local behavior of the bracket. Suppose $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ are generalized Lagrangians, such that for some fixed
$\varphi\in\mathcal{U}$, $\delta^{(1)}E(\mathcal{L}_{1})_{\varphi}[0]$ and
$\delta^{(1)}E(\mathcal{L}_{2}){\varphi}[0]$ differ only in a region outside

$\mathcal{O}\doteq\left(J^{+}(\mathrm{supp}(F))\cup
J^{-}(\mathrm{supp}(F))\right)\cap\left(J^{+}(\mathrm{supp}(G))\cup
J^{-}(\mathrm{supp}(G))\right).$ (2.64)

By the support properties of retarded and advanced propagators of Proposition
2.4.5 we have

$\left\langle
dF_{\varphi}[0],(\mathcal{G}^{+}_{\varphi,\mathcal{L}_{1}}-\mathcal{G}^{+}_{\varphi,\mathcal{L}_{2}})dG_{\varphi}[0]\right\rangle=0,$

as well as

$\left\langle
dF_{\varphi}[0],(-\mathcal{G}^{-}_{\varphi,\mathcal{L}_{1}}+\mathcal{G}^{-}_{\varphi,\mathcal{L}_{2}})dG_{\varphi}[0]\right\rangle=0.$

Taking the sum of the two we find

$\\{F,G\\}_{\mathcal{L}_{1}}-\\{F,G\\}_{\mathcal{L}_{2}}=0.$

∎

######  Theorem 2.4.10.

Let $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ $CO$-open and
$\mathcal{L}$ a generalized Lagrangian. If $F$, $G\in\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$ we have that $\\{F,G\\}_{\mathcal{L}}\in\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$ as well.

###### Proof.

By Faà di Bruno’s formula,

$\displaystyle
d^{k}\left\\{F,G\right\\}_{\mathcal{L},\varphi}[0](\vec{X}_{1},\ldots,\vec{X}_{k})$
(2.65) $\displaystyle\qquad=\sum_{(J_{1},J_{2},J_{3})\subset
P_{k}}\left\langle F^{(|J_{1}|+1)}_{\varphi}[0](\otimes_{j_{1}\in
J_{1}}\vec{X}_{j_{1}})\right.,\left.d^{(|J_{2}|)}\mathcal{G}_{\varphi}(\otimes_{j_{2}\in
J_{2}}\vec{X}_{j_{2}})G^{(|J_{3}|+1)}_{\varphi}[0](\otimes_{j_{3}\in
J_{3}}\vec{X}_{j_{3}})\right\rangle.$

while by (2.52),

$\displaystyle\begin{aligned}
d^{|J_{2}|}\mathcal{G}_{\varphi}(\vec{X}_{1},\ldots,\vec{X}_{k})=\sum_{l=1}^{k}(-1)^{l}&\sum_{\begin{subarray}{c}(I_{1},\ldots,I_{l})\\\
\in\mathcal{P}(J_{2})\end{subarray}}\sum_{p=0}^{l}\bigg{(}\bigcirc_{i=1}^{p}\mathcal{G}^{-}_{\varphi}\circ\delta^{(|I_{i}|+1)}E(\mathcal{L})\big{(}\vec{X}_{I_{i}}\big{)}\bigg{)}\\\
&\circ\mathcal{G}_{\varphi}\circ\bigg{(}\bigcirc_{i=p+1}^{l}\delta^{(|I_{i}|+1)}E(\mathcal{L})_{\varphi}\big{(}\vec{X}_{I_{i}}\big{)}\circ\mathcal{G}^{+}_{\varphi}\bigg{)},\end{aligned}$
(2.66)

where $\bigcirc_{i=1}^{p}$ stands for composition of mappings indexed by $i$
from $1$ to $p$. For the rest of the proof, we will use the integral notation
we used in (2.24) and in the proof of Proposition 2.4.5. Recall that, by
(2.42), the mapping $\delta^{(n)}E(\mathcal{L})_{\varphi}[0]$ has associated a
compactly supported integral kernel
$\mathcal{L}(1)^{(n+1)}_{\varphi}[0](x,z_{1},\ldots,z_{n})$ and its wave front
is in $N^{*}\triangle_{n+1}(M)$ by Proposition 2.4.8. Then again, we have two
general cases:
$1)$ $J_{2}=\emptyset$.
Then, letting $|J_{1}|=p$, $|J_{3}|=q=k-p$, the typical term has the form

$\displaystyle
d^{k}\left\\{F,G\right\\}_{\varphi}[0](z_{1},\ldots,z_{k})=\int_{M^{2}}f^{(p+1)}_{\varphi}[0]_{i}(x,z_{1},\ldots,z_{p}),\mathcal{G}_{\varphi}^{ij}(x,y)g^{(q+1)}_{\varphi}[0]_{j}(y,z_{p+1},\ldots,z_{k})d\mu_{g}(x,y).$
(2.67)

Suppose by contradiction that there is some
$(x_{1},\ldots,x_{k},\xi_{1},\ldots,\xi_{k})\in\mathrm{WF}(\left\\{F,G\right\\}_{\varphi}[0])$
has $(\xi_{1},\ldots,\xi_{k})\in\overline{V}^{+}_{k}(x_{1},\ldots,x_{k})$ (the
argument works similarly for
$(\xi_{1},\ldots,\xi_{k})\in\overline{V}^{-}_{k}(x_{1},\ldots,x_{k})$). Using
twice Theorem 8.2.14 in [43] in the above pairing yields

$\displaystyle\mathrm{WF}\Big{(}\big{\\{}F,G\big{\\}}^{(k)}_{\varphi}[0]\Big{)}$
$\displaystyle\qquad\subseteq\Big{\\{}(z_{1},\ldots,z_{k},\xi_{1},\ldots,\xi_{k}):\
\exists(y,\eta)\in
T^{*}M(x,z_{1},\ldots,z_{p},-\eta,\xi_{1},\ldots,\xi_{p})\in\mathrm{WF}(F^{(p+1)}_{\varphi}[0]_{i}),$
$\displaystyle\qquad\qquad(x,z_{p+1},\ldots,z_{k},\eta,\xi_{p+1},\ldots,\xi_{k})\in\mathrm{WF}\big{(}\mathcal{G}_{\varphi}^{ij}G^{(q+1)}_{\varphi}[0]_{j}\big{)}\Big{\\}}$
$\displaystyle\qquad\subset\Big{\\{}(z_{1},\ldots,z_{k},\xi_{1},\ldots,\xi_{k}):\
\exists(x,\eta),(y,\zeta)\in
T^{*}M:(x,z_{1},\ldots,z_{p},-\eta,\xi_{1},\ldots,\xi_{p})\in\mathrm{WF}\big{(}F^{(p+1)}_{\varphi}[0]_{i}\big{)}$
$\displaystyle\qquad\qquad(x,y,\eta,-\zeta)\in\mathrm{WF}(\mathcal{G}_{\varphi}^{ij}),\
(y,z_{p+1},\ldots,z_{k},\zeta,\xi_{p+1},\ldots,\xi_{k})\in\mathrm{WF}(G^{(q+1)}_{\varphi}[0]_{j})\Big{\\}}.$

So if
$(z_{1},\ldots,z_{k},\xi_{1},\ldots,\xi_{k})\in\mathrm{WF}\big{(}\left\\{F,G\right\\}^{(k)}_{\varphi}\big{)}$,
then $\exists$ $(x,\eta),(y,\zeta)\in T^{*}M$ such that

$\left\\{\begin{array}[]{ll}(x,z_{1},\ldots,z_{p},-\eta,\xi_{1},\ldots,\xi_{p})&\in\mathrm{WF}(F^{(p+1)}_{\varphi}[0]_{i}))\\\
(x,y,\eta,-\zeta)&\in\mathrm{WF}(\mathcal{G}_{\varphi}^{ij})\\\
(y,z_{p+1},\ldots,z_{k},\zeta,\xi_{p+1},\ldots,\xi_{k})&\in\mathrm{WF}(G^{(q+1)}_{\varphi}[0]_{j})\
.\end{array}\right.$

Now by Theorem 1.2.16, $\mathrm{WF}(\mathcal{G}_{\varphi}^{ij})$ contains
pairs of lightlike covectors with opposite time orientation, therefore in case
$\eta\in\overline{V}^{+}(x)$ (resp. $\eta\in\overline{V}^{-}(x)$), then
$\zeta\in\overline{V}^{+}(y)$ (resp. $\zeta\in\overline{V}^{-}(y)$) in which
case $\mathrm{WF}\big{(}G^{(q+1)}_{\varphi}[0]_{j}\big{)}$ (resp.
$\mathrm{WF}\big{(}F^{(p+1)}_{\varphi}[0]_{i}\big{)}$) does violate the
microcausality condition of Definition 2.4.6.
$2)$ $J_{2}\neq\emptyset$.
Again let $|J_{1}|=p$, $|J_{3}|=k-q$, set also, referring to (2.66),
$|I_{j}|=k_{j}$ for $j=1,\ldots,l$ so that $|J_{2}|=k_{1}+\cdots+k_{l}$.
Combining (2.65) with (2.66) with the integral kernel notation we get

$\displaystyle\\{F,G\\}^{(k)}_{\varphi}[0](z_{1},\ldots,z_{k})$
$\displaystyle=\int_{M^{k}}f^{(p+1)}_{\varphi}[0]_{i}(x,z_{1},\ldots,z_{p})\mathcal{G}^{-\
ij_{1}}_{\varphi}(x,x_{1})d^{(k_{1}+2)}\mathcal{L}_{\varphi}[0]_{j_{1}i_{1}}(x_{1},y_{1},z_{I_{1}})$
(2.68) $\displaystyle\quad\mathcal{G}^{-\
i_{1}j_{2}}_{\varphi}(y_{1},x_{2})\cdots\mathcal{G}^{-\
i_{m-1}j_{m}}_{\varphi}(y_{m-1},x_{m})d^{(k_{m}+2)}\mathcal{L}_{\varphi}[0]_{j_{m}i_{m}}(x_{m},y_{m},z_{I_{m}})$
$\displaystyle\quad\mathcal{G}_{\varphi}^{i_{m}j_{m+1}}(y_{m},x_{m+1})d^{(k_{m+1}+2)}\mathcal{L}_{\varphi}[0]_{j_{m+1}i_{m+1}}(x_{m+1},y_{m+1},z_{I_{m+1}})$
$\displaystyle\quad\mathcal{G}^{+\
i_{m+1}j_{m+2}}_{\varphi}(y_{m+1},z_{m+2})\ldots
d^{(k_{l}+2)}\mathcal{L}_{\varphi}[0]_{j_{l}i_{l}}(x_{l},y_{l},z_{I_{l}},)\mathcal{G}^{+\
i_{l}j}(y_{l},y)$ $\displaystyle
g^{(k-q+1)}_{\varphi}[0]_{j}(y,z_{q+1},\ldots,z_{k})d\mu_{g}(x,x_{1},y_{1},\ldots,x_{l},y_{l},y).$

Combining Theorem 8.2.14 in [43], Theorem 1.2.16 and Proposition 2.4.8 we can
estimate the wave front set of the integral kernel of
$\\{F,G\\}^{(k)}_{\varphi}[0]$ as all elements
$(z_{1},\ldots,z_{k},\xi_{1},\ldots,\xi_{k})\in T^{*}M^{k}$ for which there
are $(x,\eta)$, $(x_{1},\eta_{1})$, $\ldots,(x_{l}\eta_{l})$,
$(y_{1},\zeta_{1})$, $\ldots,(y_{l},\zeta_{l})$ $(y,\zeta)\in T^{*}M$ such
that

$\left\\{\begin{array}[]{ll}(x,z_{1},\ldots,z_{p},-\eta,\xi_{1},\ldots,\xi_{p})&\in\mathrm{WF}\big{(}f^{(p+1)}_{\varphi}[0]_{i}\big{)},\\\
(x,x_{1},\eta,-\eta_{1})&\in\mathrm{WF}(\mathcal{G}^{-\ ij_{1}}_{\varphi}),\\\
(x_{1},y_{1},z_{|I_{1}|},\eta_{1},-\zeta_{1},\xi_{I_{1}})&\in\mathrm{WF}\big{(}d^{(k_{1}+2)}\mathcal{L}_{\varphi}[0]_{i_{1}j_{1}}\big{)},\\\
\vdots&\vdots\\\
(y_{m-1},x_{m},\zeta_{m-1},-\eta_{m})&\in\mathrm{WF}(\mathcal{G}^{-\
i_{m-1}j_{m}}_{\varphi}),\\\
(x_{m},y_{m},z_{I_{m}},\eta_{m},-\zeta_{m},\xi_{I_{m}})&\in\mathrm{WF}\big{(}d^{(k_{m}+2)}\mathcal{L}_{\varphi}[0]_{i_{m}j_{m}}\big{)},\\\
(y_{m},x_{m+1},\zeta_{m},-\eta_{m+1})&\in\mathrm{WF}(\mathcal{G}_{\varphi}^{i_{m}j_{m+1}}),\\\
(x_{m+1},y_{m+1},z_{I_{m+1}},\eta_{m+1},-\zeta_{m+1},\xi_{I_{m+1}})&\in\mathrm{WF}\big{(}d^{(k_{m+1}+2)}\mathcal{L}_{\varphi}[0]_{i_{m+1}j_{m+1}}\big{)},\\\
(y_{m+1},x_{m+2},\zeta_{m+1},-\eta_{m+2})&\in\mathrm{WF}(\mathcal{G}^{+\
i_{m+1}j_{m+2}}_{\varphi}),\\\ \vdots&\vdots\\\
(y_{l-1},x_{l},\zeta_{l-1},-\eta_{l})&\in\mathrm{WF}(\mathcal{G}^{+\
i_{l-1}j_{l}}_{\varphi}),\\\
(x_{l},y_{l},z_{I_{l}},\eta_{l},-\zeta_{l},\xi_{I_{l}})&\in\mathrm{WF}\big{(}d^{(k_{l}+2)}\mathcal{L}_{\varphi}[0]_{i_{l}j_{l}}\big{)},\\\
(y_{l},y,\zeta_{l},-\zeta)&\in\mathrm{WF}(\mathcal{G}^{+\
i_{l}j}_{\varphi}),\\\
(y,z_{k-q+1},\ldots,z_{k},\zeta,\xi_{k-q+1},\ldots,\xi_{k})&\in\mathrm{WF}\big{(}g^{(k-q+1)}_{\varphi}[0]_{j}\big{)}.\end{array}\right.$

Suppose by contradiction, as above, that
$(z_{1},\ldots,z_{k},\xi_{1},\ldots,\xi_{k})\in\mathrm{WF}\big{(}\\{F,G\\}_{\varphi}^{(k)}\big{)}$
has $(\xi_{1},\ldots,\xi_{k})\in\overline{V}^{+}_{k}(z_{1},\ldots,z_{k})$
$\big{(}$resp.
$(\xi_{1},\ldots,\xi_{k})\in\overline{V}^{-}_{k}(z_{1},\ldots,z_{k})\big{)}$.
Then $\zeta_{m}$ and $\eta_{m+1}$ are both either lightlike future directed,
or lightlike past directed. In the first case, propagation of singularities
implies that $\zeta$ is lightlike future directed, contradicting
microcausality of $G^{(k-q+1)}_{\varphi}[0]$ (resp. $\zeta$ is lightlike past
directed, contradicting the microlocality of $G^{(k-q+1)}_{\varphi}[0]$); in
the second case, propagation of singularities implies that $\eta$ is lightlike
past directed, contradicting microcausality of $F^{(p+1)}_{\varphi}[0]$ (resp.
$\eta$ is lightlike future directed, contradicting the microlocality of
$F^{(p+1)}_{\varphi}[0]$). We remark that in the wave front set of
$\\{F,G\\}_{\varphi}^{(k)}[0]$ is the (finite) union under all possible
choices of indices for all wave front sets of the form (2.67) or (2.68), each
of which is however microcausal, implying that their finite union will be
microcausal as well. ∎

######  Theorem 2.4.11.

The mapping $(F,G)\mapsto\\{F,G\\}_{\mathcal{L}}$ defines a Lie bracket on
$\mathcal{F}_{\mu c}(B,\mathcal{U},g)$, for all $\varphi\in\mathcal{U}$
$CO$-open.

###### Proof.

Bilinearity and antisymmetry are clear from Definition 2.4.4, while Theorem
2.4.10 ensures the closure of the bracket operation. We are thus left with the
Jacobi identity:

$\\{F,\\{G,H\\}_{\mathcal{L}}\\}_{\mathcal{L}}+\\{G,\\{H,F\\}_{\mathcal{L}}\\}_{\mathcal{L}}+\\{H,\\{F,G\\}_{\mathcal{L}}\\}_{\mathcal{L}}=0.$

Using the integral kernel notation as in the above proof, we have

$\displaystyle\\{F,\\{G,H\\}_{\mathcal{L}}\\}_{\mathcal{L}}(\varphi)=\int_{M^{2}}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)\\{G,H\\}^{(1)}_{\mathcal{L}\
\varphi}[0]_{j}(y)d\mu_{g}(x,y)$
$\displaystyle=\int_{M^{4}}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)\Big{(}g^{(2)}_{\varphi}[0]_{jk}(y,z)\mathcal{G}_{\varphi}^{kl}(z,w)h^{(1)}_{\varphi}[0]_{l}(w)$
$\displaystyle\qquad+g^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{kl}_{\varphi}(z,w)h^{(2)}_{\varphi}[0]_{jl}(y,w)\Big{)}d\mu_{g}(x,y,z,w)$
$\displaystyle-\int_{M^{6}}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)\Big{(}d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y,y_{1},x_{1},)\mathcal{G}_{\varphi}^{-\
kj_{1}}(z,y_{1})g^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}_{\varphi}^{i_{1}l}(x_{1},w)h^{(1)}_{\varphi}[0]_{l}(w)$
$\displaystyle\ \ \ \ \ \ \ \ \
+d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y_{1},x_{1},y)\mathcal{G}_{\varphi}^{kj_{1}}(z,y_{1})g^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{+\
i_{1}l}_{\varphi}(x_{1},w)h^{(1)}_{\varphi}[0]_{l}(w)\Big{)}d\mu_{g}(x,y,z,w,x_{1},y_{1}).$

Summing over cyclic permutations of the first two terms yields

$\displaystyle\int_{M^{4}}$
$\displaystyle\Big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)g^{(2)}_{\varphi}[0]_{jk}(y,z)\mathcal{G}_{\varphi}^{kl}(z,w)h^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)g^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{kl}_{\varphi}(z,w)h^{(2)}_{\varphi}[0]_{jl}(y,w)}$
$\displaystyle+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)h^{(2)}_{\varphi}[0]_{jk}(y,z)\mathcal{G}_{\varphi}^{kl}(z,w)f^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}g^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)h^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{kl}_{\varphi}(z,w)f^{(2)}{\varphi}[0]_{jl}(y,w)}$
$\displaystyle+{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}h^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)f^{(2)}{\varphi}[0]_{jk}(y,z)\mathcal{G}_{\varphi}^{kl}(z,w)g^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}h^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)f^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{kl}_{\varphi}(z,w)g^{(2)}_{\varphi}[0]_{jl}(y,w)}\Big{)}d\mu_{g}(x,y,z,w)$
$\displaystyle=0,$

while for the other two,

$\displaystyle\int_{M^{6}}d\mu_{g}(x,y,z,w,x_{1},y_{1})$
$\displaystyle\Big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}x_{1}}(y,y_{1},i_{1},)\mathcal{G}_{\varphi}^{-\
kj_{1}}(z,y_{1})g^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}_{\varphi}^{i_{1}l}(x_{1},w)h^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}f^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y,y_{1},x_{1})\mathcal{G}_{\varphi}^{kj_{1}}(z,y_{1})g^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{+\
i_{1}l}_{\varphi}(x_{1},w)h^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y,y_{1},x_{1},)\mathcal{G}_{\varphi}^{-\
kj_{1}}(z,y_{1})h^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}_{\varphi}^{i_{1}l}(x_{1},w)f^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}g^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y,y_{1},x_{1})\mathcal{G}_{\varphi}^{kj_{1}}(z,y_{1})h^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{+\
i_{1}l}_{\varphi}(x_{1},w)f^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}h^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y,y_{1},x_{1},)\mathcal{G}_{\varphi}^{-\
kj_{1}}(z,y_{1})f^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}_{\varphi}^{i_{1}l}(x_{1},w)g^{(1)}_{\varphi}[0]_{l}(w)}$
$\displaystyle+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}h^{(1)}_{\varphi}[0]_{i}(x)\mathcal{G}_{\varphi}^{ij}(x,y)d^{(3)}\mathcal{L}_{\varphi}[0]_{jj_{1}i_{1}}(y,y_{1},x_{1})\mathcal{G}_{\varphi}^{kj_{1}}(z,y_{1})f^{(1)}_{\varphi}[0]_{k}(z)\mathcal{G}^{+\
i_{1}l}_{\varphi}(x_{1},w)g^{(1)}_{\varphi}[0]_{l}(w)}\Big{)}$
$\displaystyle=0.$

To make the simplifications we used the antisymmetry of the integral kernel
$\mathcal{G}_{\varphi}(x,y)$, the adjoint relation between the propagators
$\mathcal{G}^{+}(x,y)=\mathcal{G}^{-}(y,x)$ (see Lemma 2.4.3) and
$\mathcal{G}_{\varphi}^{ij}=\mathcal{G}_{\varphi}^{ji}$. ∎

###  2.5 Structure of the space of microcausal functionals

The first point of emphasis is to give a topology to $\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$. We shall proceed step by step refining our starting
definitions to better grasp the reasoning behind the choice of topology we
will be giving $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$.

The simplest guess, as well as the weakest, on $\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$ is the locally convex topology that corresponds to the
initial topology induced by the mappings

$F\rightarrow F(\varphi)\in\mathbb{R}.$

To account for smooth functionals we try the initial topology with respect to
mappings

$\displaystyle F\rightarrow F(\varphi)\in\mathbb{R},$ $\displaystyle
F\rightarrow\nabla^{k}F_{\varphi}[0]\in\Gamma^{-\infty}_{c}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right).$

This time we are leaving out all information on the wave front set which plays
a role in defining microcausal functionals. To remedy we would like to set up
the Hörmander topology on the spaces
$\Gamma^{-\infty}_{\Upsilon_{k,g}}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$,
however this is not immediately possible since $\Upsilon_{k,g}$ are open
cones, and the Hörmander topology is given to closed ones, therefore we need
the following result, whose proof can be found in Lemma 4.1 in [13],

######  Lemma 2.5.1.

Given the open cone $\Upsilon_{k}(g)$ it is always possible to find a sequence
of closed cones $\\{\mathcal{V}_{m}(k)\subset T^{*}M^{k}\\}_{m\in\mathbb{N}}$
such that $\mathcal{V}_{m}(k)\subset\mathrm{Int}(\mathcal{V}_{m+1}(k))$ and
$\cup_{m\in\mathbb{N}}\mathcal{V}_{m}(k)=\Upsilon_{k}(g)$ for all $k\geq 1$.

Then we can write

$\displaystyle\Gamma^{-\infty}_{c\
\Upsilon_{k}(g)}\left(M^{k}\leftarrow\boxtimes_{k}\varphi^{*}VB\right)=\underrightarrow{\lim_{m\in\mathbb{N}}}\Gamma^{-\infty}_{c\
\mathcal{V}_{m}(k)}\left(M^{k}\leftarrow\boxtimes_{k}\varphi^{*}VB\right).$
(2.69)

By construction of the direct limit we have mappings

$\Gamma^{-\infty}_{c\
\mathcal{V}_{m}(k)}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)\rightarrow\Gamma^{-\infty}_{c\
\Upsilon_{k}(g)}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$

where the source space has $\mathcal{V}_{m}(k)\subset T^{*}M^{k}$ as a closed
cone, so it can be given the Hörmander topology. In particular when we are
dealing with standard compactly supported distributions its topology can be
defined, see the remark after Theorem 18.1.28 in [42], to be the initial
topology with respect to the mappings

$\displaystyle F\rightarrow F(\varphi)\in\mathbb{C},$ $\displaystyle
F\rightarrow
PF\in\Gamma^{\infty}_{c}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$

where $\varphi$ is any smooth section of $B$ and $P$ any properly supported
pseudo-differential operator of order zero on the vector bundle
$\boxtimes^{k}\left(\varphi^{*}VB\right)\rightarrow M^{k}$ such that
$\mathrm{WF}(P)\cap\mathcal{V}_{m}(k)=\emptyset$. Using that Definition
18.1.32 [42], Theorem 18.1.16 [42] and Theorem 8.2.13 in [43] can be
generalized to the vector bundle case provided we use the notion (1.7) for the
wave front set of vector valued distributions; we can argue as in Corollary
4.1 of [13] that each $\Gamma^{-\infty}_{c\
\mathcal{V}_{m}(k)}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$
becomes a Hausdorff topological space. By Theorem 2.1.15 since the base
manifold $M$ is separable and the fibers are finite dimensional vector spaces,
hence nuclear999We say that a Hausdorff locally convex space E is nuclear if
given any other locally convex space F we have $E\otimes_{\pi}F\simeq
E\otimes_{\epsilon}F$, where the two are the tensor product space respectively
endowed with the quotient topology and with the canonical topology associated
to the space of continuous bilinear mappings $:E^{\prime}_{\sigma}\times
F^{\prime}_{\sigma}\rightarrow\mathbb{R}$ equipped with the topology of
uniform convergence on products of equicontinuous subsets of $E^{\prime}$ and
$F^{\prime}$. For more detail see either [60] or [69]. and Fréchet, we find
that
$\Gamma^{\infty}_{c}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$
is a nuclear limit-Fréchet space, it is Hausdorff, thus each

$\Gamma^{-\infty}_{c\
\mathcal{V}_{m}(k)}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$

is nuclear as well. Finally by Porposition 50.1 pp. 514 in [69] the direct
limit topology on

$\Gamma^{-\infty}_{c\
\Upsilon_{k,g}}\left(M^{k}\leftarrow\boxtimes^{k}(\varphi^{*}VB)\right)$

is nuclear for all $k$ (and also Hausdorff). We have therefore proved:

######  Lemma 2.5.2.

The direct limit topology on $\Gamma^{-\infty}_{c\
\Upsilon_{k,g}}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$
induced as a direct limit topology of the spaces $\Gamma^{-\infty}_{c\
\mathcal{V}_{m}(k)}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$
with the Hörmander topology is a Hausdorff nuclear space.

Finally we can induce on $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ a topology by

######  Theorem 2.5.3.

Given the set $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$, consider the mappings

$\mathcal{F}_{\mu c}(B,\mathcal{U},g)\ni F\mapsto F(\varphi)\in\mathbb{R},$
(2.70) $\mathcal{F}_{\mu c}(B,\mathcal{U},g)\ni
F\mapsto\nabla^{k}F_{\varphi}[0]\in\Gamma^{-\infty}_{c\
\Upsilon_{k,g}}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right),$
(2.71)

and the related initial topology on $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$.
Then $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ is a nuclear locally convex
topological space with a Poisson *-algebra with respect to the Peierls bracket
of some microlocal generalized Lagrangian $\mathcal{L}$.

###### Proof.

The nuclearity follows from the stability of nuclear spaces under projective
limit topology, see Proposition 50.1 pp. 514 in [69], so using the nuclearity
of both $\Gamma^{-\infty}_{c\
\Upsilon_{k,g}}\left(M^{k}\leftarrow\boxtimes^{k}\left(\varphi^{*}VB\right)\right)$
(via Lemma 2.5.2) and $\mathbb{R}$ (trivially) we have our claim. The Peierls
bracket is well defined by Theorem 2.4.10 and satisfies the Jacobi identity
due to Theorem 2.4.11, so we only have to show the Leibniz rule for the
bracket, that is

$\\{F,GH\\}_{\mathcal{L}}=G\\{F,H\\}_{\mathcal{L}}+\\{F,G\\}_{\mathcal{L}}H.$

However this follows once we show that the product $F,G\mapsto F\cdot G$ with
$(F\cdot G)(\varphi)=F(\varphi)G(\varphi)$ is closed in $\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$ and then use $d(F\cdot
G)_{\varphi}[0]=dF_{\varphi}[0]G(\varphi)+F(\varphi)dG_{\varphi}[0]$. The
latter is a consequence of the definition of derivation (i.e. the standard
Leibniz rule), so we are left with showing the former: we compute

$d^{k}(F\cdot
G)_{\varphi}[0](\vec{X}_{1},\ldots,\vec{X}_{k})=\sum_{\sigma\in\mathcal{P}(1,\ldots,k)}\sum_{l=0}^{k}d^{l}F_{\varphi}[0](\vec{X}_{\sigma(1)},\ldots,\vec{X}_{\sigma(l)})d^{k-l}G_{\varphi}[0](\vec{X}_{\sigma(k-l+1)},\ldots,\vec{X}_{\sigma(k)}),$

where $\mathcal{P}(1,\ldots,k)$ is the set of permutations of
$\\{1,\ldots,k\\}$. For each of those terms using Theorem 8.2.9 in [43] we
have

$\displaystyle\mathrm{WF}(F^{(l)}_{\varphi}[0]G^{(k-l)}_{\varphi}[0])$
$\displaystyle\subset\mathrm{WF}(F^{(l)}_{\varphi}[0])\times\mathrm{WF}(G^{(k-l)}_{\varphi}[0])$
$\displaystyle\quad\bigcup\mathrm{WF}(G^{(k-l)}_{\varphi}[0])\times\left(\mathrm{supp}(G^{(k-l)}_{\varphi}[0])\times\\{{0}\\}\right)$
$\displaystyle\quad\bigcup\left(\mathrm{supp}(F^{(l)}_{\varphi}[0])\times\\{{0}\\}\right)\times\mathrm{WF}(G^{(k-l)}_{\varphi}[0]),$

and therefore microcausality is met. ∎

Note that closed linear subspaces of $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$
are nuclear as well (see Proposition 50.1 in [69]), so $\mathcal{F}_{\mu
loc}(B,\mathcal{U},g)$ is a Hausdorff nuclear space. The space
$\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ can be given a structure of a
$C^{\infty}$-ring, more precisely

######  Proposition 2.5.4.

If $F_{1},\ldots,F_{n}$ $\in\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ and $\psi\in
V\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ is smooth, then
$\psi(F_{1},\ldots,F_{n})\in\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ and

$\mathrm{supp}(\psi(F_{1},\ldots,F_{n}))\subset\bigcup_{i=1}^{n}\mathrm{supp}(F_{i}).$

###### Proof.

First we check the support properties. Suppose that
$x\notin\cup_{i=1}^{n}\mathrm{supp}(F_{i})$, we can find an open neighborhood
$V$ of $x$ for which given any $\varphi\in\mathcal{U}$ and any
$\vec{X}\in\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}VB)$ having
$\mathrm{supp}(\vec{X})\subset V$ implies $(F_{i}\circ
u_{\varphi})(t\vec{X})=(F_{i}\circ u_{\varphi})(0)$ for all $t$ in a suitable
neighborhood of $0\in\mathbb{R}$. Then $\psi\big{(}(F_{1}\circ
u_{\varphi})(t\vec{X}),\ldots,(F_{n}\circ
u_{\varphi})(t\vec{X})\big{)}=\psi\big{(}(F_{1}\circ
u_{\varphi})(0),\ldots,(F_{n}\circ u_{\varphi})(0)\big{)}$ as well giving
$x\notin\mathrm{supp}(\psi\circ(F_{1},\ldots,F_{n}))$. We immediately see that
the composition is Bastiani smooth, so consider its $k$th derivative expressed
via Faà di Bruno’s formula:

$\displaystyle
d^{k}\psi(F_{1},\ldots,F_{1})_{\varphi}[0](\vec{X}_{1},\ldots,\vec{X}_{k})$
$\displaystyle=\sum_{\begin{subarray}{c}(J_{1},\ldots,J_{n})\\\
\in\mathcal{P}(1,\ldots,k)\end{subarray}}\frac{\partial^{k}\psi(F_{1}(\varphi),\ldots,F_{n}(\varphi))}{\partial
z^{J_{1}+\ldots+J_{n}}}\left(F^{(|J_{1}|)}_{1\
\varphi}[0](\vec{X}_{j_{1,1}},\ldots,\vec{X}_{j_{|J_{1}|,1}})\cdot\ldots\cdot
F^{(|J_{n}|)}_{n\
\varphi}[0](\vec{X}_{j_{1,n}},\ldots,\vec{X}_{j_{|J_{n}|,n}})\right)$

where $\mathcal{P}(1,\ldots,k)$ denotes the set of partitions of
$\\{1,\ldots,k\\}$ and $z\in\mathbb{R}^{n}$. Since $\psi$ is smooth the only
contribution to the wavefront set of the composition is the product of
functional derivative in the above sum, for which Theorem 8.2.9 in [43] gives

$\displaystyle\mathrm{WF}$
$\displaystyle\left(F_{1}^{(|J_{1}|)},\ldots,F_{n}^{(|J_{n}|)}\right)$
$\displaystyle\subset$ $\displaystyle\quad\mathrm{WF}\left(F_{1\
\varphi}^{(|J_{1}|)}\right)\times\ldots\times\mathrm{WF}\left(F_{n\
\varphi}^{(|J_{n}|)}\right)$
$\displaystyle\quad\bigcup\mathrm{supp}\left(F_{1\
\varphi}^{(|J_{1}|)}\right)\times\\{\vec{0}\\}^{|J_{1}|}\times\mathrm{WF}\left(F_{2\
\varphi}^{(|J_{2}|)}\right)\times\ldots\times\mathrm{WF}\left(F_{n\
\varphi}^{(|J_{n}|)}\right)$ $\displaystyle\quad\ldots$
$\displaystyle\quad\bigcup\mathrm{WF}\left(F_{1\
\varphi}^{(|J_{1}|)}\right)\times\ldots\times\mathrm{WF}\left(F_{n-1\
\varphi}^{(|J_{n-1}|)}\right)\times\mathrm{supp}\left(F_{n\
\varphi}^{(|J_{n}|)}\right)\times\\{\vec{0}\\}^{|J_{n}|}$
$\displaystyle\quad\ldots$ $\displaystyle\quad\bigcup\mathrm{supp}\left(F_{1\
\varphi}^{(|J_{1}|)}\right)\times\\{\vec{0}\\}^{|J_{1}|}\times\ldots\times\mathrm{supp}\left(F_{n-1\
\varphi}^{(|J_{n-1}|)}\right)\times\\{\vec{0}\\}^{|J_{n-1}|}\times\mathrm{WF}\left(F_{n\
\varphi}^{(|J_{n}|)}\right).$

We clearly have that if an element of
$\mathrm{\mathrm{WF}}\left(F_{1}^{(|J_{1}|)},\ldots,F_{n}^{(|J_{n}|)}\right)$
was contained in either $\overline{V}^{k}_{+,g}$ or $\overline{V}^{k}_{-,g}$
then at least one of the initial functional cannot be microcausal. ∎

Going through the same calculation for the proof of Proposition 2.5.4 we get
the expression for the Peierls bracket of this composition:

$\\{\psi(F_{1},\ldots,F_{n}),G\\}_{\mathcal{L}}=\sum_{j=1}^{n}\left(\frac{\partial\psi}{\partial
z^{j}}(F_{1},\ldots,F_{n})\\{F_{j},G\\}_{\mathcal{L}}\right).$ (2.72)

With the topology of Theorem 2.5.3 the space of microcausal functionals lacks
sequential continuity. Consider as an example the simpler case where
$B=M\times\mathbb{R}$, then choose
$F:\varphi\in\mathcal{U}\mapsto\int_{M}\varphi(x)\omega$ for some smooth
compactly supported $m$-form $\omega$ over $M$, also let $\\{f_{n}\\}$ be a
sequence of smooth functions $:\mathbb{R}\rightarrow[0,1]$ supported in
$[-2,2]$ and converging pointwise to the characteristic function of $[-1,1]$,
$\chi_{[-1,1]}$, then sequences of derivatives of $f_{n}$ all converge
punctually to the zero function on $\mathbb{R}$. If we define
$F_{n}(\varphi)=f_{n}\circ F(\varphi)$, then pointwise
$F_{n}(\varphi)\rightarrow\chi_{[-1,1]}\circ F(\varphi)$ but

$F^{(k)}_{n\
\varphi}[0](\psi_{1},\ldots,\psi_{k})=f^{(k)}_{n}(F(\varphi))\int_{M}\psi_{1}(x)\omega(x)\ldots\int_{M}\psi_{k}(x)\omega(x)$

converges pointwise to the zero functional in the microcausal topology,
however $\chi_{[-1,1]}\circ F(\cdot)$ is not even continuous, and definitely
not microcausal. Thus the completion of this topology might be somewhat
uncontrolled.

######  Remark.

To define a topology that is both nuclear and has a well behaving sequential
completion, we induce, as in [13], the strong convenient topology. We start
with the following observations:

  * •

let $\mathcal{U}$ be a $CO$-open subset of $\Gamma^{\infty}(M\leftarrow B)$,
then the smooth curves $\gamma:I\subseteq\mathbb{R}\to\mathcal{U}$ are
precisely the conveniently smooth curves from $I$ to $\mathcal{U}$;

  * •

by construction each $\mathcal{U}$ can be decomposed as the union of CO-open
subsets $\sqcup_{\varphi\in\mathcal{U}}\mathcal{U}\cap\mathcal{V}_{\varphi}$,
each of which is topologically isomorphic to an open subset of
$\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}VB)$ with the LF topology;

  * •

combining Corollary 2.1.9 we see that each smooth curve
$\gamma:I\to\mathcal{U}$ is just valued in some $\mathcal{V}_{\varphi}$ for
some $\varphi\in\mathcal{U}$.

  * •

if $\varphi_{0}\in\mathcal{U}$ is fixed, we can can consider the ultralocal
chart representation of any functional $F\in\mathcal{F}_{c}(\mathcal{U},B)$ as
$F_{\varphi_{0}}:\mathcal{U}_{0}\equiv
u_{\varphi_{0}}(\mathcal{U}\cap\mathcal{U}_{\varphi_{0}})\to\mathbb{R}$. Using
the fact that the functional has compact support we can extend it to
$\widetilde{F}_{\varphi_{0}}=F_{\varphi_{0}}\circ
i_{\chi}:\widetilde{\mathcal{U}}_{0}\subset\Gamma^{\infty}(M\leftarrow\varphi_{0}^{*}VB)\to\mathbb{R}$
where $\chi\in C^{\infty}_{c}(M)$ has $\chi|_{\mathrm{supp(F)}}\equiv 1$,
$i_{\chi}(\vec{X})=\chi\vec{X}$ and
$\widetilde{\mathcal{U}}_{0}=i_{\chi}^{-1}(\mathcal{U}_{0})$ is $CO$-open.

  * •

if we equip $\Gamma^{\infty}(M\leftarrow\varphi_{0}^{*}VB)$ with the $CO$-open
topology, then it becomes a Fréchet space (the compact-open topology is
equivalent to the topology of uniform convergence on compact subsets), thus,
on $\widetilde{\mathcal{U}}_{0}$, the notions of Bastiani smoothness and
convenient smoothness do coincide by Proposition 1.3.23.

Next we note that, by (1.16), that we can write

$C^{\infty}(\widetilde{\mathcal{U}}_{0},\mathbb{R})=\lim_{\begin{subarray}{c}\longleftarrow\\\
\widetilde{\gamma}\in
C^{\infty}(\mathbb{R},\widetilde{\mathcal{U}}_{0})\end{subarray}}C^{\infty}(\mathbb{R},\mathbb{R})=\bigg{\\{}\\{\widetilde{F}\\}_{\widetilde{\gamma}}\in\prod_{\widetilde{\gamma}\in
C^{\infty}(\mathbb{R},\widetilde{\mathcal{U}}_{0})}C^{\infty}(\mathbb{R},\mathbb{R}):\widetilde{F}_{\widetilde{\gamma}}\circ\kappa=\widetilde{F}_{\widetilde{\gamma}\circ\kappa}\bigg{\\}}$

where the inverse limit is taken with respect to the pre-order
$\gamma\leq\gamma^{\prime}$ if and only if there is $\kappa\in
C^{\infty}(\mathbb{R},\mathbb{R})$ with $\gamma=\gamma^{\prime}\circ\kappa$.
Moreover we induce on $C^{\infty}(\widetilde{\mathcal{U}}_{0},\mathbb{R})$ the
initial topology from the Fréchet space topology on
$C^{\infty}(\mathbb{R},\mathbb{R})$ through the pullbacks $\gamma^{*}$. This
is a nuclear and sequentially complete topology (see the discussion in remark
4.3 pp. 55 on [13]); finally, since $\mathcal{F}_{c}(B,\mathcal{U}_{0})$ is
closed in $C^{\infty}(\widetilde{\mathcal{U}}_{0},\mathbb{R})$, nuclearity and
completeness are inherited in the quotient topology. This space is even a
locally convex topological vector space with the seminorms

$\sup_{\begin{subarray}{c}t\in[a,b]\subset\mathbb{R}\\\ \gamma\in
C^{\infty}(\mathbb{R},\mathcal{U}_{0})\\\
(\vec{X}_{1},\ldots,\vec{X}_{k})\in\mathcal{B}\subset\boxtimes^{k}\Gamma^{\infty}(M\leftarrow\varphi^{*}VB)\end{subarray}}\big{|}\nabla^{(k)}F[\gamma(t)](\vec{X}_{1},\ldots,\vec{X}_{k})\big{|},$
(2.73)

where
$\mathcal{B}\subset\boxtimes^{k}\Gamma^{\infty}(M\leftarrow\varphi^{*}VB)$ is
a closed, bounded subset. Notice that due to $F$ having compact support, then
we can evaluate $\nabla^{(k)}F[\gamma(t)](\vec{X}_{1},\ldots,\vec{X}_{k})$ by
taking a properly chosen cutoff $\chi\in C^{\infty}_{c}(M)$ and calculating
$\nabla^{(k)}F[\gamma(t)](\chi\vec{X}_{1},\ldots,\chi\vec{X}_{k})$.

We can then induce a topology $\tau_{\mathrm{sc}}$, which we call the strong
convenient topology, in $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ with (2.73) in
place of (2.70), together with the seminorms (2.71). Then $\tau_{\mathrm{sc}}$
will enjoy the following properties:

  * (a)

it remains a nuclear locally convex space topology;

  * (b)

will have a well controlled and nuclear101010See Proposition 5.3.1 in [60].
completion, i.e. its completion amounts to the topology induced from the
completion of the spaces $\Gamma^{-\infty}_{c\
\Upsilon_{k,g}}(M\leftarrow\varphi^{*}VB)$;

  * (c)

the Poisson *-algebra and $C^{\infty}$-ring operations are continuous and
remain such when passing to the completion described in (b), thanks to the
result of [7].

Before the next result, let us recall some notions from [50]. A topological
space $(X,\tau)$ is Lindelöf if given any open cover of $X$ there is a
countable open subcover, it is separable if it admits a countable dense
subset, and it is second countable if it admits a countable basis for the
topology.
Let $X$ be a Hausdorff locally convex topological space, possibly infinite
dimensional, and take $S\subset C(X,\mathbb{R})$, a subalgebra. We say that
$X$ is $S$-normal if $\forall A_{0},A_{1}$ closed disjoint subsets of $X$
there is some $f\in S$ such that $f|_{A_{i}}=i$, while we say it is
$S$-regular if for any neighborhood $U$ of a point $x$ there exists a function
$f\in S$ such that $f(x)=1$ and $\mathrm{supp}(f)\subset U$. A $S$-partition
of unity is a family $\\{\psi_{j}\\}_{j\in J}$ of mappings
$S\ni\psi_{j}:X\rightarrow\mathbb{R}$ with

  * $(i)$

$\psi_{j}(x)\geq 0$ for all $j\in J$ and $x\in X$;

  * $(ii)$

the set $\\{\mathrm{supp}(\psi_{j}):j\in J\\}$ is a locally finite covering of
$X$,

  * $(iii)$

$\sum_{j\in J}\psi_{j}(x)=1$ for all $x\in X$.

When $X$ admits such partition we say it is $S$-paracompact.

######  Proposition 2.5.5.

The following facts hold true:

  * $(i)$

Given any $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ $CO$-open and any
$\varphi_{0}\in\mathcal{U}$ there is some $F\in\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$ such that $F(\varphi_{0})=1$, $0\leq F|_{\mathcal{U}}\leq
1$ and $F|_{\Gamma^{\infty}(M\leftarrow
B)\backslash\mathcal{U}_{\varphi_{0}}}=0$, i.e. $\mathcal{U}$ is
$\mathcal{F}_{\mu caus}(B,\varphi_{0},g)$-regular.

  * $(ii)$

Any $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ $CO$-open admits
locally finite partitions of unity belonging to $\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$.

  * $(iii)$

Given any $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ $CO$-open, the
algebra $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$ separates the points of
$\mathcal{U}$, that is if $\varphi_{1}\neq\varphi_{2}$ there is a microcausal
functional $F$ that has $F(\varphi_{1})\neq F(\varphi_{2})$.

###### Proof.

To show $(i)$ take the chart $(\mathcal{U}_{\varphi_{0}},u_{\varphi_{0}})$ and
consider the open subset $\mathcal{U}\cap\mathcal{U}_{\varphi_{0}}$, fix some
compact $K\subset M$ and some
$\omega\in\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}VB^{\prime}\otimes\Lambda_{m}(M))$
with $\mathrm{supp}(\omega)\subset K$, then we can define a functional

$G_{\omega}:\mathcal{U}\cap\mathcal{U}_{\varphi_{0}}\ni\varphi\mapsto
G_{\omega}(\varphi)=\int_{M}\omega(u_{\varphi_{0}}(\varphi)).$

Denote now by $G$ the functional with $G(\varphi)\doteq G_{\omega}\circ
u_{\varphi_{0}}^{-1}(u_{\varphi_{0}}(\varphi))$. By construction
$G(\varphi_{0})=0$. Let now
$\mathcal{W}=\\{\varphi\in\mathcal{U}\cap\mathcal{U}_{0}:G(\varphi)<\epsilon^{2}\\}$
for some constant $\epsilon$, then if $\chi:\mathbb{R}\rightarrow\mathbb{R}$
is a smooth function supported in $[-1,1]$ with $0\leq\chi|_{[-1,1]}\leq 1$
and $\chi|_{[-1/2,1/2]}\equiv 1$, consider the new functional
$F=\chi\circ(\frac{1}{\epsilon^{2}}G)$, since $G$ is microlocal (and thus
microcausal by Proposition 2.4.8) and $\chi$ is smooth, by Propositions 2.5.4
$F$ is microcausal. Outside $\mathcal{W}$, $F$ is identically zero so we can
smoothly extend it to zero over the rest of $\mathcal{U}$ to a new functional
which we denote always by $F$ that has the required properties. We first show
that $(ii)$ holds for $U_{\varphi}$. Using the chart
$(U_{\varphi},u_{\varphi})$, we can identify $U_{\varphi}$ with an open subset
of $\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}VB)$. If we show that
$U_{\varphi}$ is Lindelöf and is $\mathcal{F}_{\mu
c}(B,\varphi_{0},g)$-regular, then we can conclude via Theorem 16.10 pp. 171
of [50]. $\mathcal{F}_{\mu c}(B,\mathcal{U},g)$-regularity was point $(i)$
while the Lindelöf property follows from Theorem 2.1.15. Now we observe that
any $\mathcal{U}$ can be obtained as the disjoint union of subsets
$\mathcal{V}_{\varphi_{0}}\doteq\\{\psi\in\mathcal{U}:\mathrm{supp}_{\varphi_{0}}(\psi)\
\mathrm{is}\ \mathrm{compact}\\}$. Each of this is Lindelöf and metrizable by
Theorem 2.1.15, so given the open cover
$\\{\mathcal{U}_{\varphi}\\}_{\varphi\in\mathcal{V}_{\varphi_{0}}}$, we can
extract a locally finite subcover where each elements admits a partition of
unity and then construct a partition of unity for the whole
$\mathcal{V}_{\varphi_{0}}$. The fact that
$\mathcal{U}=\sqcup\mathcal{V}_{\varphi_{0}}$ implies that the final partition
of unity is the union of all others. Finally for $(iii)$ just take
$\mathcal{U}_{\varphi_{1}}$, $\mathcal{U}_{\varphi_{2}}$ and $F$ as in $(i)$
constructed as follows: if $\varphi_{2}\in\mathcal{U}_{1}$ we choose
$\epsilon<G(\varphi_{2})$ for which $F(\varphi_{1})\neq F(\varphi_{2})$, if
not then any $\epsilon>0$ does the job. ∎

######  Definition 2.5.6.

Let $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ be $CO$-open and
$\mathcal{L}$ a generalized microlocal Lagrangian. We define the on-shell
ideal associated to $\mathcal{L}$ as the subspace
$\mathcal{I}_{\mathcal{L}}(B,\mathcal{U},g)\subset\mathcal{F}_{\mu
c}(B,\mathcal{U},g)$ whose microcausal functionals are of the form

$F(\varphi)=\vec{X}_{\varphi}\left(E(\mathcal{L})_{\varphi}[0]\right)$ (2.74)

with $X:\mathcal{U}\rightarrow
T\mathcal{U}:\varphi\mapsto(\varphi,\vec{X}_{\varphi})$ is a smooth vector
field.

With our usual integral kernel notation we can also write (2.74) as

$F(\varphi)=\int_{M}\vec{X}_{\varphi}^{i}(x)E(\mathcal{L})_{\varphi}[0]_{i}(x)d\mu_{g}(x).$
(2.75)

We stress that functionals of the form (2.74) are those which can be seen as
the derivation of the Euler-Lagrange derivative by kinematical vector fields
over $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$.

######  Proposition 2.5.7.

$\mathcal{I}_{\mathcal{L}}(B,\mathcal{U},g)$ is a Poisson $*$-ideal of
$\mathcal{F}_{\mu c}(B,\mathcal{U},g)$.

###### Proof.

Clearly is $F\in\mathcal{I}_{\mathcal{L}}(B,\mathcal{U},g)$ then also
$\overline{F}$ is. If $G\in\mathcal{F}_{\mu c}(B,\mathcal{U},g)$, $G\cdot
F(\varphi)=G(\varphi)X(\varphi)\left(E(\mathcal{L})_{\varphi}[0]\right)$ but
then $X^{\prime}=G\cdot X\in\mathfrak{X}(T\mathcal{U})$ as well, then $G\cdot
F$ is in the ideal and is associated to the new vector field $X^{\prime}$.
Finally we have to show that if
$F\in\mathcal{I}_{\mathcal{L}}(B,\mathcal{U},g)$ then also
$\\{F,G\\}_{\mathcal{L}}\in\mathcal{I}_{\mathcal{L}}(B,\mathcal{U},g)$. Fix
$\varphi\in\mathcal{U}$, $\vec{Y}_{\varphi}\in T_{\varphi}\mathcal{U}$; by the
chain rule

$dF_{\varphi}[0](\vec{Y}_{\varphi})=\int_{M}\left[\vec{X}^{(1)}_{\varphi}[0]^{i}(\vec{Y}_{\varphi})\left(E(\mathcal{L})_{\varphi}[0]_{i}\right)+\vec{X}_{\varphi}^{i}\left(E^{(1)}(\mathcal{L})_{\varphi}[0]_{i}(\vec{Y}_{\varphi})\right)\right]d\mu_{g}$

and

$\displaystyle\\{F,G\\}_{\mathcal{L}}(\varphi)$ $\displaystyle=\left\langle
dF_{\varphi}[0],\mathcal{G}_{\varphi}dG_{\varphi}[0]\right\rangle$
$\displaystyle=\int_{M}\left[\vec{X}^{(1)}_{\varphi}[0]^{i}_{j}\left(\mathcal{G}_{\varphi}^{jk}g^{(1)}_{\varphi}[0]_{k}\right)\left(E(\mathcal{L})_{\varphi}[0]_{i}\right)+\vec{X}_{\varphi}^{i}\left(E^{(1)}(\mathcal{L})_{\varphi}[0]_{ij}\left(\mathcal{G}_{\varphi}^{jk}g^{(1)}_{\varphi}[0]_{k}\right)\right)\right]d\mu_{g}$
$\displaystyle=\int_{M}\left\\{\left[\vec{X}^{(1)}_{\varphi}[0]^{i}_{j}\left(\mathcal{G}_{\varphi}^{jk}g^{(1)}_{\varphi}[0]_{k}\right)\right]\left(E(\mathcal{L})_{\varphi}[0]_{i}\right)\right\\}d\mu_{g}$

where we used that $\mathcal{G}_{\varphi}$ associates to its argument a
solution of the linearized equations. Defining
$\varphi\mapsto\vec{Z}_{\varphi}=\vec{X}^{(1)}_{\varphi}[0]^{i}_{j}\left(\mathcal{G}_{\varphi}^{jk}g^{(1)}_{\varphi}[0]_{k}\right)\partial_{i}\in\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}VB)$
yields a smooth mapping (by smoothness of $X$, the functional $G$ and the
propagator $\mathcal{G}_{\varphi}$) defining the desired vector field. ∎

######  Definition 2.5.8.

Let $\mathcal{U}\subset\Gamma^{\infty}(M\leftarrow B)$ be $CO$-open and
$\mathcal{L}$ a generalized microlocal Lagrangian. We define the on-shell
algebra on $\mathcal{U}$ associated to $\mathcal{L}$ as the quotient

$\mathcal{F}_{\mathcal{L}}(B,\mathcal{U},g)\doteq\mathcal{F}_{\mu
c}(B,\mathcal{U},g)/\mathcal{I}_{\mathcal{L}}(B,\mathcal{U},g).$ (2.76)

This accounts for the algebra of observable once the condition
$E(\mathcal{L})_{\varphi}[0]=0$ has been imposed on $\mathcal{U}$.

###  2.6 Examples

####  2.6.1 Wave maps

Finally we introduce, as an example of physical theory, wave maps. The
configuration bundle, is $C=M\times N$, where $M$ is an $m$ dimensional
Lorentzian manifold and $N$ an $n$ dimensional manifold equipped with a
Riemannian metric $h$. The space of sections is canonically isomorphic to
$C^{\infty}(M,N)$, the latter possess a differentiable structure induced by
the atlas
$\big{(}\mathcal{U}_{\varphi},u_{\varphi},\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}TN)\big{)}$,
where $u_{\varphi}$ has the exact same form (2.9), with the only difference
being that the sections are $N$-valued mappings, thus $\exp$ can be taken as
the exponential function induced by a Riemannian metric $h$ on $N$. The
generalized Lagrangian for wave maps is

$\mathcal{L}_{\mathrm{WM}}(f)(\varphi)=\frac{1}{2}\int_{M}f(x)\mathrm{Trace}(g^{-1}\circ(\varphi^{*}h))(x)d\mu_{g}(x);$
(2.77)

obtained by integration of the standard geometric Lagrangian
$\lambda=\frac{1}{2}g^{\mu\nu}h_{ij}(\varphi)\varphi^{i}_{\mu}\varphi^{j}_{\nu}d\mu_{g}$
smeared with a test function $f\in C^{\infty}_{c}(M)$. Computing the first
functional derivative, as per (2.39), we get the associated E-L equations,
which written in jets coordinates reads

$h_{ij}g^{\mu\nu}\left(\varphi^{i}_{\mu\nu}+\\{h\\}^{i}_{kl}\varphi^{k}_{\mu}\varphi^{l}_{\nu}-\\{g\\}^{\lambda}_{\mu\nu}\varphi^{i}_{\lambda}\right)=0,$
(2.78)

where we denoted by $\\{h\\}$, $\\{g\\}$ the coefficients of the linear
connection associated to $h$ and $g$ respectively. Computation of the second
derivative of (2.77) yields

$\displaystyle\delta^{(1)}E(\mathcal{L}_{WM})_{\varphi}[0]$
$\displaystyle:\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}TN)\times\Gamma^{\infty}_{c}(M\leftarrow\varphi^{*}TN)\rightarrow\mathbb{R}$
(2.79) $\displaystyle(\vec{X},\vec{Y})\mapsto$
$\displaystyle\int_{M}\frac{1}{2}\left[g^{\mu\nu}(x)h_{ij}(\varphi(x))\nabla_{\mu}{X}^{i}(x)\nabla_{\nu}{Y}^{j}(x)+A^{\mu}_{ij}(\varphi(x))\left(\nabla_{\mu}{X}^{i}(x){Y}^{j}(x)+\nabla_{\mu}{Y}^{i}(x){X}^{j}(x)\right)\right.$
$\displaystyle\ \ \ \ \
+\left.B_{ij}(\varphi(x)){X}^{i}(x){Y}^{j}(x)\right]d\mu_{g}(x).$

Where we choose $f\equiv 1$ in a neighborhood of
$\mathrm{supp}(\vec{X})\cup\mathrm{supp}(\vec{X})$ as done before. One can
show that the coefficients $A^{\mu}_{ij}$ always vanish and

$\begin{split}&\delta^{(1)}E(\mathcal{L}_{WM})_{\varphi}[0](\vec{X},\vec{Y})\\\
&\qquad=\int_{M}\frac{1}{2}\big{(}g^{\mu\nu}(x)h_{ij}(\varphi(x))\nabla_{\mu}{X}^{i}(x)\nabla_{\nu}Y^{j}(x)+R^{k}_{ilj}(\varphi(x))p^{\alpha}_{k}(\varphi(x))\varphi^{l}_{\alpha}(x){X}^{i}(x)Y^{j}(x)\big{)}d\mu_{g}(x)\end{split}$

where $R$ are the components of the Riemann tensor of the Riemannian metric
$h$, and $p^{\alpha}_{k}\doteq\frac{\partial\lambda}{\partial y^{k}_{\alpha}}$
is the conjugate momenta of the Lagrangian $\lambda$. It is therefore evident
that the induced differential operator $D_{\varphi}$ can be expressed locally
as

$D_{\varphi}(\vec{X})(x)=\big{(}g^{\mu\nu}(x)h_{ij}(\varphi(x))\nabla_{\mu\nu}\vec{X}^{i}(x)+R^{k}_{ilj}(\varphi(x))p^{\alpha}_{k}(\varphi(x))\varphi^{l}_{\alpha}(x){X}^{i}(x)\big{)}dy^{j}\big{|}_{\varphi(x)}\
.$ (2.80)

Its principal symbol is clearly

$\sigma_{2}(D_{\varphi})=\frac{1}{2}g^{\mu\nu}\frac{\partial}{\partial
x^{\nu}}\frac{\partial}{\partial x^{\mu}}\otimes id_{\varphi^{*}TN}.$

Theorem 1.2.16 then ensures the existence of the advanced and retarded
propagators for Wave Maps $\mathcal{G}^{\pm}_{WM}[\varphi]$. Their difference
defines the causal propagator and consequently the Peierls bracket as in
Definition 2.4.4. The results of Sections 2.3 and 2.4 do apply to wave maps:
it is therefore possible to obtain a $*$-Poisson algebra generated by
microcausal functionals $\mathcal{F}_{\mu{c}}(M\times N,g)$ which enjoys all
the properties collected throughout Section 2.4.

####  2.6.2 Scalar field theories

We finish this chapter by translating some of the results obtained above in
case where $B=M\times\mathbb{R}$. We will use those results in the following
chapters. We notice at first that the manifold stricture of
$C^{\infty}(M,\mathbb{R})$ is generated by charts
$\\{(\mathcal{U}_{\varphi},u_{\varphi},C^{\infty}_{c}(M))\\}$ where

$\displaystyle\mathcal{U}_{\varphi}=\varphi+C^{\infty}_{c}(M);$ (2.81)
$\displaystyle u_{\varphi}(\psi)=\psi-\varphi.$

Notice that the standard Euclidean metric on $\mathbb{R}$ has a globally
defined exponential $\exp:T\mathbb{R}\to\mathbb{R}\times\mathbb{R}$, then
$\mathcal{U}_{\varphi}$ becomes the largest possible subset isomorphic to a
copy of the modelling vector space. This enlargement of the chart open set
yields a ultralocal chart independent result for the characterization of
microlocal functionals in Proposition 2.3.11, therefore if
$F:C^{\infty}(M,\mathbb{R})$ is microlocal and $F^{(1)}$ is locally
bornological, then

$F(\varphi)=F(\varphi_{0})=\int_{M}j^{r}_{x}\varphi^{*}\theta_{F,\varphi_{0}}$
(2.82)

for all $\varphi$ having $\varphi-\varphi_{0}\in C^{\infty}_{c}(M)$. We also
remark that $C^{\infty}(M,\mathbb{R})$ can be seen as a disjoint union as

$C^{\infty}(M,\mathbb{R})=\bigsqcup_{\varphi\in
C^{\infty}(M,\mathbb{R})/\sim}\varphi+C^{\infty}_{c}(M)$

where $\sim$ is the equivalence relation defined by
$\varphi_{1}\sim\varphi_{2}$ if and only if their difference
$\varphi_{1}-\varphi_{2}$ is compactly supported. In this sense, (2.82) is a
ultralocal chart independent object. Finally, due to Proposition 2.3.12, we
also get that the

$F(\varphi)=\int_{M}j^{r}_{x}\varphi^{*}\theta_{F}$

for a globally defined $\theta\in\Omega^{n}(M)$.

In the sequel the dynamics will be that given by the Klein-Gordon Lagrangian:

$\mathcal{L}_{f}:C^{\infty}(M,\mathbb{R})\ni\varphi\mapsto\int_{M}f(x)\big{(}g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi+m^{2}\varphi^{2}+\kappa
R(g)\varphi^{2}\big{)}(x)d\mu_{g}(x).$ (2.83)

The latter generates the Klein-Gordon equations

$P(\varphi)\equiv
d\mathcal{L}_{1}[\varphi](x)=\big{(}g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi+m^{2}\varphi+\kappa
R(g)\big{)}\varphi.$ (2.84)

In particular the constants $m$, $\kappa$ are called the mass and the coupling
constant of the theory. We also mention a special kind of functionals, i.e.
those generating Wick powers, we denote them by

$\phi^{k}_{(M,g)}(f):C^{\infty}(M,\mathbb{R})\ni\varphi\mapsto\int_{M}f(x)\varphi^{k}(x)d\mu_{g}(x).$
(2.85)

It it easy to show that both (2.83), (2.85) are microlocal and are generalized
Lagrangians according to Definition 2.3.13.

Finally, let us show some other results for the scalar case which we will
extensively use in the following parts. In the scalar case the strong
convenient topology is generated by seminorms

$p_{[a,b],\gamma,\mathcal{B}}(F)\doteq\sup_{\begin{subarray}{c}t\in[a,b]\subset\mathbb{R}\\\
\gamma\in C^{\infty}(\mathbb{R},\mathcal{U})\\\
(\psi_{1},\ldots,\psi_{k})\in\mathcal{B}\end{subarray}}\left|d^{k}F[\gamma(t)](\psi_{1},\ldots,\psi_{k})\right|,$
(2.86)
$p_{\varphi,\Upsilon_{k,g},q,\chi}(F)\doteq||F^{(k)[\varphi]}||_{q,\chi,V},$
(2.87)

where $\gamma:\mathbb{R}\rightarrow\mathcal{U}$ is a smooth curve valued in a
$c^{\infty}$-open subset of $C^{\infty}(M)$, $\mathcal{B}$ is a closed and
bounded subset in $C^{\infty}(M)^{k}\simeq C^{\infty}(M^{k})$,
$q\in\mathbb{N}$, $\chi\in C^{\infty}_{c}(M^{k})$, $V\subset T^{*}M^{k}$ with
$\Upsilon_{k,g}\cap\mathrm{supp}(\chi)\times V=\emptyset$ are Hörmander
seminorms in the direct limit topology of Lemma 2.5.2. Then we have the
following technical result:

######  Lemma 2.6.1.

$\mathcal{F}_{reg}(M,g)\subset\mathcal{F}_{\mu c}(M,g)$ is sequentially dense
in the strong convenient topology.

###### Proof.

Let $F\in\mathcal{F}_{\mu c}(M,g)$, we show that there exists a sequence
$\\{F_{n}\\}\subset\mathcal{F}_{reg}(M,g)$ such that $F_{n}\rightarrow F$ in
the strong convenient topology of $\mathcal{F}_{\mu c}(M,g)$. First we search
for a sequence of mollifiers strongly converging to the identity mapping. We
shall use [19, Theorem 12 pp.68], which guarantees the existence of continuous
linear mappings $S_{n}:C^{\infty}(M)\rightarrow C^{\infty}(M)$ such that
$S_{n}\rightarrow id_{C^{\infty}(M)}$ uniformly over bounded subsets of
$C^{\infty}(M)$. Suppose that $F\in\mathcal{F}_{\mu c}(M,h)$ has support
inside a compact $K\subset M$, and let $\chi\in C^{\infty}_{c}(M)$ with
$\chi|_{K}=1$. Set $F_{n}(\varphi)=F(S_{n}(\chi_{n}\varphi))$, since each
$S_{n}$ enlarges the support at most in a $1/n$ neighborhood of $K$,
$F(S_{n}(\chi_{n}\varphi))=F(S_{n}(\varphi))$; by the Schwartz kernel theorem
we can represent $S_{n}(\chi\varphi)$ as

$S_{n}(\chi\varphi)(x)=\int_{M}S_{n}(x,y)\chi(y)\varphi(y)d\mu_{g}(y),$

where $S_{n}\in C^{\infty}(M^{2})$ for each $n\in\mathbb{N}$. To simplify the
notation, we shall write $S_{n}(\varphi)$ in place of $S_{n}(\chi\varphi)$. We
claim that $\mathrm{WF}(d^{k}F_{n}[\varphi])=\emptyset$ $\forall
n,k\in\mathbb{N}$. To wit, by linearity of each $S_{n}$,
$dS_{n}[\varphi](\psi)=S_{n}(\psi)$, $d^{k}S_{n}[\varphi]=0$ $\forall k\geq
2$; thus the only non trivial term in Faà di Bruno’s formula is

$d^{k}F_{n}[\varphi](\psi_{1},\ldots,\psi_{k})=d^{k}F[S_{n}(\varphi)](dS_{n}[\varphi](\psi_{1}),\ldots,dS_{n}[\varphi](\psi_{k}))=d^{k}F[S_{n}(\varphi)](S_{n}(\psi_{1}),\ldots,S_{n}(\psi_{k})).$

Its associated kernel, $d^{k}F_{n}[S_{n}(\varphi)](x_{1},\ldots,x_{k})$, will
have the form

$\int_{M^{k}}d^{k}F[S_{n}(\varphi)](y_{1},\ldots,y_{k})S_{n}(x_{1},y_{1})\cdots
S_{n}(x_{k},y_{k})d\mu_{g}(y_{1},\ldots,y_{k}).$

By smoothness of each $S_{n}$, singularities of $d^{k}F[S_{n}(\varphi)]$ are
suppressed and $d^{k}F_{n}[S_{n}(\varphi)](x_{1},\ldots,x_{k})$ becomes
smooth, therefore $F_{n}\in\mathcal{F}_{reg}(M)$. Finally we have to show
convergence in the strong convenient topology, which is generated by the
family of seminorms in (2.86) and (2.87). Since we are working with regular
functionals, the second seminorms are trivial and all we have to check is
convergence with respect to the first. In practice, chosen any seminorm and
$\epsilon>0$, we have to show that there is some $n_{0}$ for which $n>n_{0}$
implies

$p_{\gamma,[a,b],B,k}(F_{n}-F)<\epsilon.$

Thus we estimate

$\displaystyle p_{\gamma,[a,b],B,k}(F_{n}-F)$
$\displaystyle=\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left|d^{k}F[S_{n}(\gamma(t))](S_{n}(\psi_{1}),\ldots,S_{n}(\psi_{k}))-d^{k}F[\gamma(t)](\psi_{1},\ldots,\psi_{k})\right|$
$\displaystyle\leq$ $\displaystyle\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left|d^{k}F[S_{n}(\gamma(t))](S_{n}(\psi_{1}),\ldots,S_{n}(\psi_{k}))-d^{k}F[S_{n}(\gamma(t)](\psi_{1},\ldots,\psi_{k})\right|$
$\displaystyle\quad+\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left|d^{k}F[S_{n}(\gamma(t))](\psi_{1},\ldots,\psi_{k})-d^{k}F[\gamma(t)](\psi_{1},\ldots,\psi_{k})\right|$
$\displaystyle\quad\leq\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left|d^{k}F[S_{n}(\gamma(t))](S_{n}(\psi_{1})-\psi_{1},\ldots,S_{n}(\psi_{k})-\psi_{k})\right|$
$\displaystyle\quad+\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left|p_{S_{n}(\gamma(t)),\psi_{1},\ldots,\psi_{k}}(F)-p_{\gamma(t),\psi_{1},\ldots,\psi_{k}}(F)\right|.$

$F:C^{\infty}(M)\rightarrow\mathbb{R}$ is convenient smooth, hence Bastiani
smooth due to the Fréchet space topology in $C^{\infty}(M)$; as a result all
its derivatives are jointly continuous, and we can further estimate

$\displaystyle\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left\\{Cq_{K,l}(S_{n}(\gamma(t)))\prod_{i=1}^{k}q_{K,l}(S_{n}(\psi_{i})-\psi_{i}))\right\\}$
$\displaystyle+\sup_{\begin{subarray}{c}t\in[a,b]\\\
(\psi_{1},\ldots,\psi_{k})\in
B\end{subarray}}\left\\{C^{\prime}(q_{K,l}(S_{n}(\gamma(t))-q_{K,l}(\gamma(t)))\prod_{i=1}^{k}q_{K,l}(\psi_{i}))\right\\}$
$\displaystyle\leq$ $\displaystyle
C\sup_{t\in[a,b]}\left\\{q_{K,l}(S_{n}(\gamma(t)))\right\\}\sup_{(\psi_{1},\ldots,\psi_{k})\in
B}\left\\{\prod_{i=1}^{k}q_{K,l}(S_{n}(\psi_{i})-\psi_{i}))\right\\}$
$\displaystyle+C^{\prime}\sup_{t\in[a,b]}\left\\{q_{K,l}(S_{n}(\gamma(t))-\gamma(t))\right\\}\sup_{(\psi_{1},\ldots,\psi_{k})\in
B}\left\\{\prod_{i=1}^{k}q_{K,l}(\psi_{i}))\right\\}$

where $q_{K,l}$ are the seminorms of the Fréchet topology on $C^{\infty}(M)$.
The terms $\sup_{t\in[a,b]}\left\\{q_{K,l}(S_{n}(\gamma(t)))\right\\}$,
$\sup_{(\psi_{1},\ldots,\psi_{k})\in
B}\left\\{\prod_{i=1}^{k}q_{K,l}(\psi_{i}))\right\\}$ are bounded by constants
$D$ and $D^{\prime}$ respectively, due to continuity of the seminorms;
moreover, since $S_{n}$ converges uniformly on bounded subsets of
$C^{\infty}(M)$, there is $n_{0}\in\mathbb{N}$, having
$\sup_{(\psi_{1},\ldots,\psi_{k})\in
B}\left\\{q_{K,l}(S_{n}(\psi_{i})-\psi_{i}))\right\\}<\frac{\epsilon}{2CD}$
and $\sup_{\psi\in
B^{\prime}\supset\gamma([a,b])}\left\\{q_{K,l}(S_{n}(\psi)-\psi)\right\\}<\frac{\epsilon}{2C^{\prime}D^{\prime}}$,
finally establishing our claim. Notice that those estimates do not depend on
the cut-off $\chi$ for choosing its support big enough, we can directly assume
$\chi\equiv 1$. ∎

##  Chapter 3  Functional formalism in quantum field theory: Wick powers

In this chapter we are transitioning from classical free field theories to
quantum free field theories. First we review the definitions and properties of
natural bundles: they are particular types of bundles which, broadly speaking,
have a canonical way for lifting local diffeomorphisms of the base manifold to
local automorphisms of the bundle. This notion is very common for one of the
first bundles encountered in differential geometry is the tangent bundle,
which has the aforementioned property, in particular given a local
diffeomorphism we can associate its Jacobian matrix, and this is all is needed
to create the trivializations of the tangent space. We stress this lifting of
diffeomorphism (as well as the tangent mapping) has functorial origin, we will
then see (Theorem 3.1.5, Theorem 3.1.4), that natural bundles can be
equivalently described by a functor or through a geometrical construction. The
reason we are interested in such objects is that they provide a framework for
handling the geometric parameters of the theory which, in the case of Klein-
Gordon theory in curved spacetime, will result in defining the natural bundle
of background geometries (see (3.8) and Definition 3.1.7) whose sections
$h=(g,m,\kappa)$ represents the metric, the mass and the coupling constant to
gravity. The importance of such bundles is crucial for classical field
theories for it gives a natural way of treating covariant functions and forms
(for example see Theorem 3.1.9); however, it can be used in the quantum
setting as well and will provide the technical framework for classifying
ambiguities in the definition of Wick powers (c.f. Theorem 3.3.10).

The starting point for the quantum theory are the classical results obtained
in Chapter 2, then we apply deformation quantization to the Poisson algebra of
microcausal functionals $(\mathcal{F}_{\mu c},\\{\ ,\ \\}_{(M,h)})$
($h=(g,m,\kappa)$ is a background geometry), which consists in deforming the
classical product of functionals to a $\star$-product. This is however, not
directly possible for microcausal functionals; to find a way around we
restrict the algebra to regular functionals and define there the $\star$
product. Introducing the notion of Hadamard state and using the
characterization [63, Theorem 5.1] we show that it is possible to define a
$\star_{H}$-product on $\mathcal{F}_{\mu c}$, where $H$ is a Hadamard
parametrix. The issue here is that this $\star$ product is not uniquely
defined for any two parametrix $H,\ H^{\prime}$ differ by a smooth function
$d\in C^{\infty}(M\times M)$. Using the density of regular functionals into
microcausal, we extend the regular $\star$ product to a $\star$ product in the
abstract algebra of microcausal functionals $\mathfrak{A}_{\mu c}$ with the
property that whenever $H$ is the symmetric part of a Hadamard state, we have
an isomorphism of $\star$ algebras

$\alpha_{H}:(\mathfrak{A}_{\mu c},\star)\to(\mathcal{F}_{\mu c},\star_{H}).$

The next step is the definition of Wick powers (see Definition 3.3.9). As
mentioned earlier, in [39], the characterization of uniqueness has been
established using the rather strong analytical dependence of Wick powers on
the background geometry. This condition was weakened in [46] to a milder one
which was enough for using the Peetre-Slovák Theorem and classify ambiguities
in the definition of Wick powers. For the sake of thoroughness, we will state
and prove the results obtained in [46, 45] for scalar field theories. We
stress that, differently from [39, 46], we are not using on-shell algebras,
since functional formalism is general enough to avoid working in this special
condition. Several of the requirements of Wick powers were therefore
generalized in Definition 3.3.9.

###  3.1 Natural bundles

######  Definition 3.1.1.

A natural bundle is a covariant functor,
$\mathfrak{N}:\mathfrak{Man}\rightarrow\mathfrak{FBndl}$, from the category of
manifold and local diffeomorphisms to the category of fiber bundles and local
fibered morphisms, satisfying the following properties:

  * $(i)$

$\mathfrak{N}(M)\equiv(B,\pi,M,F)$ is a fiber bundle over $M$;

  * $(ii)$

if $i:U\hookrightarrow M$ is the inclusion of a submanifold, then
$\mathfrak{N}(i):\mathfrak{N}(U)\rightarrow\mathfrak{N}(M)$ is the inclusion
of the sub-bundle $\mathfrak{N}(U)\equiv(\pi^{-1}(U),\pi\mid_{U},U,F)$ into
the fiber bundle $\mathfrak{N}(M)$;

  * $(iii)$

If $\Sigma$ is a parameter manifold and $\phi:\Sigma\times M\rightarrow N$ is
a smooth family of local diffeomorphisms depending on the parameter
$s\in\Sigma$, then
$\mathfrak{N}(\phi):\Sigma\times\mathfrak{N}(M)\rightarrow\mathfrak{N}(N)$ is
smooth.

It is now easy to see that the tangent bundle $(TM,\tau,M,\mathbb{R}^{n})$ is
an example of natural bundle, in fact $T$ is easily seen to be a covariant
functor, $(i)$ is clear, $(ii)$ follows once we realize that
$i:U\hookrightarrow$ is a local diffeomorphism that is locally the identity,
therefore $Ti:TU\rightarrow TM$ will be locally the identity as well,
therefore forming a sub-bundle. Finally to show $(iii)$ note that the fiber
mapping $T\phi_{s}:TM\rightarrow TN$, where $s\in\Sigma$ is a parameter, will
involve products and sums of coefficients of the Jacobian of $\phi_{s}$,
$J^{\mu}_{\nu}(s)$, which is jointly smooth.

We now attempt to give a classification of the possible types of natural
bundles. In order to do so, we will take, at first sight, a somewhat strange
path which, in the end, will prove of great practical utility. To start
consider open sets $U$, $V$ of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$
respectively each of which contains the origin, let

$J^{k}_{0}(U,V)_{0}\doteq\\{\alpha\in C^{\infty}(U,V)\ :\ \alpha(0)=0\\}/\sim$
$J^{k}_{0}(\mathbb{R}^{n},\mathbb{R}^{m})_{0}\doteq\\{\alpha\in
C^{\infty}(\mathbb{R}^{n},\mathbb{R}^{m})\ :\ \alpha(0)=0\\}/\sim$

where the equivalence relation is given by $\alpha\sim\alpha^{\prime}$ if and
only if $j^{k}_{0}\alpha=j^{k}_{0}\alpha^{\prime}$. Using local coordinates
$(U,x^{i})$ and $(V,y^{a})$ we can write

$j^{k}_{0}\alpha=\alpha^{a}|_{0}+\frac{\partial\alpha^{a}}{\partial
x^{i}}\Big{|}_{0}x^{i}+\ldots+\frac{1}{k!}\frac{\partial^{k}\alpha^{a}}{\partial
x^{i_{1}}\ldots\partial x^{i_{k}}}\Big{|}_{0}x^{i_{1}}\ldots
x^{i_{k}}\equiv\alpha^{a}_{i}x^{i}+\ldots+\frac{1}{k!}\alpha^{a}_{i_{1}\ldots
i_{k}}x^{i_{1}}\ldots x^{i_{k}}$

therefore we can make this space a
$n\left(\binom{n+k}{n}-1\right)$-dimansional differentiable manifold structure
whose coordinates are of the form
$\left(\alpha^{a}_{i},\ldots\alpha^{a}_{i_{1}\ldots i_{k}}\right)$. Moreover,
provided $s<k$, we have smooth projections

$\pi^{k}_{s}:J^{k}_{0}(U,V)_{0}\rightarrow
J^{s}_{0}(U,V)_{0}:(\alpha^{a}_{i},\ldots\alpha^{a}_{i_{1}\ldots
i_{k}})\mapsto(\alpha^{a}_{i},\ldots\alpha^{a}_{i_{1}\ldots i_{s}})$

Another important property of those spaces is covariance for the jet functor
$J^{k}_{0}(\cdot,\cdot)_{0}$: let $\alpha\in J^{k}_{0}(U,V)_{0}$, $\beta\in
J^{k}_{0}(V,W)_{0}$ with $U$, $V$, $W$ open subsets of $\mathbb{R}^{n}$,
$\mathbb{R}^{m}$, $\mathbb{R}^{p}$ respectively, then
$\beta\circ\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{p}$ is well defined and
the image of $0\in\mathbb{R}^{n}$ is $0\in\mathbb{R}^{p}$, moreover by Faà di
Bruno’s formula $[\beta\circ\alpha]\in J^{k}_{0}(U,W)_{0}$. As an example, for
$k=2$ we have that locally $[\beta\circ\alpha]$ has the form

$(\beta^{a}_{j}\alpha^{j}_{i},\beta^{a}_{ij}\alpha^{i}_{k}\alpha^{j}_{l}+\beta^{a}_{i}\alpha^{i}_{kl}).$

Suppose that $m=n$, and that instead of generic mappings we only consider
those that are invertible on a neighborhood of the origin. We shall denote
this space as

$GL^{k}(n,\mathbb{R})\doteq J^{k}_{0}(U,V)^{(\mathrm{inv})}_{0}.$

Not only the latter space is a manifold but it is a Lie group as well: the
inversion map is the one associating the local inverse, while the
multiplication map is given by the composition of mappings; both operations
are made by sums and products of monomials, and, as a consequence, are smooth.
$GL^{k}(n,\mathbb{R})$ is then usually referred as the $k$th order jet group
of $GL(n,\mathbb{R})$, since, when $k=1$, we can readily identify the former
with with the general linear group of $\mathbb{R}^{n}$.

Now we go one step further, let $e:U\subset\mathbb{R}^{n}\rightarrow M$ be a
local diffeomorphism in a neighborhood of the origin $0\in\mathbb{R}^{n}$,
calculating its $k$th order jet extension in the origin yields

$(j^{k}_{0}e)^{\mu}=e^{\mu}(0)+e^{\mu}_{i}x^{i}+\ldots+e^{\mu}_{i_{1}\ldots
i_{k}}x^{i_{1}}\ldots x^{i_{k}}.$

Similarly to what we did before, let $J^{k}_{0}(U,M)^{(\mathrm{inv})}$ the
$k$th order jet space of invertible mappings with the structure of a fiber
bundle over $M$ of dimension $n\binom{n+k}{n}$, which we denote by $L^{k}(M)$.
Fibered coordinates are of the form
$(e^{\mu},e^{\mu}_{i}\ldots,e^{\mu}_{i_{1}\ldots i_{k}})$, and coordinates
transformations on $M$ with $x^{\prime\mu}=x^{\prime\mu}(x)$, induce
transformations on $L^{k}(M)$ of the form

$\left\\{\begin{array}[]{@{}l@{}}e^{\prime\mu}\equiv x^{\prime\mu}(x),\\\
e^{\prime\mu}_{i}=J^{\mu}_{\alpha}(x)e^{\alpha}_{i},\\\
e^{\prime\mu}_{ij}=J^{\mu}_{\alpha}(x)e^{\alpha}_{ij}+J^{\mu}_{\alpha\beta}(x)e^{\alpha}_{i}e^{\beta}_{j},\\\
\ldots\\\ e^{\prime\mu}_{i_{1}\ldots
i_{k}}x^{i_{1}}=J^{\mu}_{\alpha}e^{\alpha}_{i_{1}\ldots
i_{k}}x^{i_{1}}+\ldots+J^{\mu}_{\alpha_{1}\ldots\alpha_{k}}e^{\alpha_{1}}_{i_{1}}\ldots
e^{\alpha_{k}}_{i_{k}}.\end{array}\right.\,$ (3.1)

######  Lemma 3.1.2.

$L^{k}(M)\to M$ is a principal fiber bundle, called the $k$th order frame
bundle, with structure group $GL^{k}(n,\mathbb{R})$.

###### Proof.

We shall verify that the trivializations induced by (3.1) define
trivializations as per Definition 1.2.4. The projection
$\pi:L^{k}(M)\rightarrow M:j^{k}_{0}e\mapsto e(0)$ is a well defined smooth
mapping, consider some atlas $\\{U_{a},x^{\mu}\\}_{a\in A}$ of $M$, and define
trivializations

$t_{(a)}:L^{k}(M)\rightarrow M\times
GL^{k}(n,\mathbb{R}):j^{k}_{0}e\mapsto(e(0),j^{k}_{0}\alpha),$ (3.2)

where $j^{k}_{0}\alpha\in GL^{k}(n,\mathbb{R})$ is the unique element in the
equivalent class whose $k$th order jet expansion in $0\in\mathbb{R}^{n}$ is

$\alpha(0)=e^{\mu}_{i}x^{i}+\ldots+e^{\mu}_{i_{1}\ldots i_{k}}x^{i_{1}}\ldots
x^{i_{k}}+R_{k+1}(x).$

Since the mappings $t_{(a)}$ involve only polynomial terms, they are smooth.
Transition mappings associates to trivializations $t_{(a)}$ and $t_{(b)}$
takes the form

$\displaystyle t_{(ab)}:$ $\displaystyle U_{\alpha\beta}\times
GL^{k}(n,\mathbb{R})\rightarrow U_{\alpha\beta}\times GL^{k}(n,\mathbb{R})$
(3.3) $\displaystyle j^{k}_{0}(x\circ e)\equiv(e(0),j^{k}_{0}\alpha)\mapsto
j^{k}_{0}(x^{\prime}\circ e)\equiv((x^{\prime}\circ
e)(0),j^{k}_{0}(J\circ\alpha)),$

which specified in coordinates reads
$(e^{\mu},e^{\mu}_{i}\ldots,e^{\mu}_{i_{1}\ldots
i_{k}})\mapsto(e^{\prime\mu},e^{\prime\mu}_{i}\ldots,e^{\prime\mu}_{i_{1}\ldots
i_{k}})$ and the relation between the two is given by (3.1). In other words we
are identifying the transition mapping as a left action of the element
$j^{k}J\in GL^{k}(n,\mathbb{R})$, given by the Jacobian its subsequent
derivatives, on the element $j^{k}_{0}\alpha$ of the typical fiber. All of
(3.2), (3.3) are smooth in the respective domains since are polynomials in the
respective variables. ∎

Since all principal bundles come equipped with a global right action, we
explicitly write down the one of $GL^{k}(n,\mathbb{R})$ on $L^{k}(M)$:

$L^{k}(M)\times GL^{k}(n,\mathbb{R})\rightarrow
L^{k}(M):(j^{k}_{0}e,j^{k}_{0}\beta)\mapsto j^{k}_{0}(e\circ\beta),$ (3.4)

which locally, for $k=2$, reads

$(e^{\mu},e^{\mu}_{a},e^{\mu}_{ab},\alpha^{a}_{i},\alpha^{a}_{ij})\mapsto(e^{\mu},e^{\mu}_{a}\alpha^{a}_{i},e^{\mu}_{ab}\alpha^{a}_{i}\alpha^{b}_{j}+e^{\mu}_{a}\alpha^{a}_{ij}).$

The reason why $L^{k}(M)$ is called the $k$th order frame bundle is that, when
$k=1$, it becomes the usual frame bundle111The frame bundle of $M$, denoted
$L(M)$ is the set $\cup_{x\in M}\\{(e_{x})\\}$, where $\\{e_{x}\\}$ is the set
of basis of the tangent space $T_{x}M$. If a set of coordinates
$(U_{a},\\{x^{\mu}\\})$ are chosen on $M$, one can induce trivializations on
$L(M)$ by setting  $t_{a}(e_{x})=(x^{\mu},(e_{x})^{\mu}_{a}),$ where
$(e_{x})^{\mu}_{a}\in GL(n,\mathbb{R})$ is the matrix associated to the change
of basis $\partial_{\mu}\to(e_{x})_{a}$.. It is customary to introduce the
tangent bundle $TM$ of $M$ as as the space of kinematic tangent vectors of
$M$, it can, however, also be seen as the bundle associated to the frame
bundle $L(M)$ with the left action
$\lambda:GL(n,\mathbb{R})\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ that
it is induced by left multiplication of a matrix on a vector. Define a bundle

$L(M)\times_{\lambda}\mathbb{R}^{n}\doteq
L(M)\times\mathbb{R}^{n}/\sim_{\lambda},$

where the equivalence relation is given by $(p,v)\sim(p^{\prime},v^{\prime})$
if there is some $A\in GL(n,\mathbb{R})$ such that $p^{\prime}=R_{A}(p)$ and
$v^{\prime}=\lambda_{A}(v)$. Coordinates are of the form $(x^{\mu},v^{\mu})$,
and if we change them on $M$ via $x^{\prime}=x^{\prime}(x)$ we have induced a
transformation $v^{\prime\mu}=J^{\mu}_{\nu}v^{\nu}$. By (ii) in Proposition
1.2.2 $L(M)\times_{\lambda}\mathbb{R}^{n}\simeq TM$.

The case of the tangent bundle presented above points to the emergence of a
certain pattern: bundles associated with frame bundles are natural. In order
to prove it we shall need some intermediate results.

######  Lemma 3.1.3.

$L^{k}:M\mapsto L^{k}(M)$ is a functor from the category of manifolds and
local diffeomorphisms to the category of fiber bundles and local fibered
morphisms.

###### Proof.

From Lemma 3.1.2, we have that for each manifold $M$ (resp. $U\subset M$),
$L^{k}(M)$ (resp. $L^{k}(U)\subset L^{k}(M)$) is a principal fiber bundle.
Suppose that $\psi\in\mathrm{Diff}_{{loc}}(M)$, then there are open subsets
$U$,$V$ of $M$ such that $\psi:U\rightarrow V$ is a diffeomorphism. Set

$L^{k}(\varphi):L^{k}(U)\rightarrow L^{k}(V):j^{k}_{0}e\mapsto
j^{k}_{0}(\psi\circ e).$

By functoriality and covariance of the jet prolongations
$j^{k}_{0}(\varphi\circ e)=j^{k}_{e(0)}\varphi\circ j^{k}_{0}e$, yielding a
commutative diagram:

$L^{k}(U)$$L^{k}(V)$$U$$V$$L^{k}(\varphi)$$j^{k}_{0}e$$j^{k}_{0}(\varphi\circ
e)$$\varphi$

In addition, $L^{k}(\psi)\left(R_{j^{k}_{0}\alpha}j^{k}_{0}(\psi\circ
e)\right)=j^{k}_{0}(\psi\circ
e\circ\alpha)=R_{j^{k}_{0}\alpha}\left(L^{k}(\psi)j^{k}_{0}e\right)$, implying
that $L^{k}(\psi)$ is even a local principal isomorphism. ∎

Given a natural bundle functor $\mathfrak{N}$, we say that it has finite order
$k$ if, given any $M\in\mathrm{Ob}(\mathfrak{Man})$,
$\phi,\psi\in\mathrm{Diff}_{\mathrm{loc}}(M)$ with
$j^{k}_{x}\phi=j^{k}_{x}\psi$ in some $x\in M$, we have

$\mathfrak{N}(\phi)(\pi^{-1}(x))\simeq\mathfrak{N}(\psi)(\pi^{-1}(x))$

where as usual $\pi$ is the projection of the fiber bundle $\mathfrak{N}(M)$.

######  Theorem 3.1.4.

Let $\mathfrak{N}:\mathfrak{Man}\rightarrow\mathfrak{FBndl}$ be a natural
functor of finite order $k$, then for all $M\in\mathrm{Ob}(\mathfrak{Man})$ we
have a canonical isomorphism

$\mathfrak{N}(M)\simeq L^{k}(M)\times_{\lambda}S$

where $S\doteq\pi_{M}^{-1}(0)$ is the standard fiber of the natural bundle
$\mathfrak{N}(\mathbb{R}^{n})$.

###### Proof.

Denote by $\pi_{M}:\mathfrak{N}(M)\to M$ the bundle projection. We start by
writing down the action of the group $GL^{k}(n)$ on $S$. Consider two
$n$-dimensional manifolds $M$, $N$ and the mapping

$J^{k}(M,N)^{(\mathrm{inv})}\times_{M}\mathfrak{N}(M)\rightarrow\mathfrak{N}(N):\big{[}j^{k}_{x}f,b\big{]}\mapsto\mathfrak{N}(f)(b),$

where as usual the space $J^{k}(M,N)^{(\mathrm{inv})}$ denotes the space of
$k$th order jets of local diffeomorphisms and $\big{[}j^{k}_{x}f,b\big{]}$ any
representative of the equivalence relation
$\big{[}j^{k}_{x}f,b\big{]}\sim_{M}\big{[}j^{k}_{x^{\prime}}f^{\prime},b^{\prime}\big{]}$
if $x=x^{\prime},\ \pi_{M}(b)=\pi_{M}(b^{\prime})$. From here on we work
locally on the chart $U_{a}\subseteq M$ and assume $M=N$,
$U_{a}=\mathbb{R}^{n}$; in which case the above mapping takes the form

$\displaystyle
J^{k}(U_{a},U_{a})^{(\mathrm{inv})}\times_{U_{a}}\mathfrak{N}(U_{a})\equiv
J^{k}(U_{a},U_{a})^{(\mathrm{inv})}\times
S\rightarrow\mathfrak{N}(U_{a}):(j^{k}_{x}\alpha,y)\mapsto(x,\mathfrak{N}(\alpha)(y))\equiv(x,\tilde{\alpha}(y)),$

where we are trivializing points $b$ of $\mathfrak{N}(U_{a})$ as $(x,y)$.
Choosing in particular $x=0$, and composing with the fiber projection yields a
mapping

$\lambda:GL^{k}(n,\mathbb{R})\times S\rightarrow
S:(j^{k}_{0}\alpha,y)\rightarrow\tilde{\alpha}(y).$ (3.5)

This is a left action since
$\lambda_{j^{k}_{0}\alpha}\circ\lambda_{j^{k}_{0}\beta}(y)=\mathfrak{N}(\alpha)\circ\mathfrak{N}(\beta)(y)=\mathfrak{N}(\alpha\circ\beta)(y)=\lambda_{\alpha\circ\beta}(y)$.
Then consider the mapping

$L^{k}(M)\times_{\lambda}S\ni(j^{k}_{x}e,y)\mapsto(x,\mathfrak{N}(\alpha)(y))\in\mathfrak{N}(M),$
(3.6)

where $j^{k}_{x}e=(x,j^{k}_{0}\alpha)$ for some $j^{k}_{0}\alpha\in
GL^{k}(n,\mathbb{R})$. This is well defined for if we take any other
$\alpha^{\prime}$ with $j^{k}_{0}\alpha^{\prime}=j^{k}_{0}\alpha$, by
finiteness of the order of $\mathfrak{N}$,
$\mathfrak{N}(\alpha)=\mathfrak{N}(\alpha^{\prime})$. Finally this is an
isomorphism since choosing $j^{k}_{x}\overline{e}=(x,j^{k}_{0}\alpha^{-1})$,
and using covariance of the jet and natural functor, we show that (3.6) is an
isomorphism of bundles projecting over $id_{M}$. ∎

Theorem 3.1.4 can be framed into a more categorical setting: consider
therefore the mapping $\lambda:GL^{k}(m)\times S\rightarrow S$ canonically
defined by (3.5), then we induce a covariant functor

$\displaystyle N:$
$\displaystyle\mathfrak{PB}(GL^{k}(m))\rightarrow\mathfrak{FBndl}$ (3.7)
$\displaystyle P\mapsto P\times_{\lambda}S$ $\displaystyle\Psi\mapsto
L(\Psi)\equiv[\Psi,id_{S}]_{\lambda}$

where $\mathfrak{PB}(G)$ is the category of principal bundles with group $G$
and local principal bundle isomorphisms, and the action of $L(\Psi)$ on a
point $[p,y]_{\lambda}\in P\times_{\lambda}S$ is understood as
$[\Psi(p),y]_{\lambda}$. Combining this with Lemma 3.1.2 we see that we can
construct a functor $\mathfrak{L}\doteq N\circ
L^{k}:\mathfrak{Man}\rightarrow\mathfrak{FBndl}$.

######  Theorem 3.1.5.

There is a bijective correspondence between the set of $k$th order natural
bundles on $n$-dimensional manifolds and the associated $S$-bundle to a
principal $GL^{k}(n,\mathbb{R})$-bundle. Moreover this correspondence takes
the form of natural transformation between functors.

###### Proof.

Let $\mathfrak{N}$ a natural bundle functor, fixed
$M\in\mathrm{Ob}(\mathfrak{Man})$, we can consider (up to diffeomorphisms) the
standard fiber $S\simeq\pi^{-1}(x)$, where $\mathfrak{N}(M)=(B,\pi,M,S)$ is
the natural bundle in question. Call
$q_{j^{k}_{0}e}:L^{k}(M)\times_{\lambda}S\mid_{j^{k}_{0}e}\rightarrow\mathfrak{N}(M)|_{x=e(0)}$
the isomorphism introduced in Theorem 3.1.4, and set

$\chi_{M}:\mathfrak{L}(M)\rightarrow\mathfrak{N}(M):(j^{k}_{0}e,y)\mapsto\mathfrak{N}(e)(y).$

Suppose that $\psi\in\mathrm{Diff}_{(\mathrm{loc})}(M)$, then

$\mathfrak{N}(\psi)\circ\chi_{M}([j^{k}_{0}e,y]_{\lambda})=\mathfrak{N}(\psi)\circ\mathfrak{N}(e)(y)=\mathfrak{N}(\psi\circ
e)(y)=\chi_{N}([\psi\circ
e,y]_{\lambda})=\chi_{N}\circ\mathfrak{L}(\psi)([j^{k}_{0}e,y]_{\lambda}).$

Equivalently we showed that the following diagram is commutative:

$\mathfrak{L}(M)$$\mathfrak{N}(M)$$\mathfrak{L}(N)$$\mathfrak{N}(N)$$\chi_{M}$$\mathfrak{L}(\psi)$$\mathfrak{N}(\psi)$$\chi_{N}$

Therefore $\chi:\mathfrak{L}\Rightarrow\mathfrak{N}$ is a natural
transformation. ∎

A worthy example of a natural bundle is the bundle of linear connections
$\mathrm{Conn}(M)$ of $M$. The latter is defined as the associated bundle to
$L^{2}(M)$ via the action

$\lambda:GL^{2}(n)\times\mathbb{R}^{n^{3}}\rightarrow\mathbb{R}^{n^{3}}:(J^{\alpha}_{\beta},J^{\alpha}_{\beta\gamma},\Gamma^{\alpha}_{\beta\gamma})\mapsto
J^{\alpha}_{\beta}\left(\Gamma^{\beta}_{\mu\nu}\bar{J}^{\mu}_{\gamma}\bar{J}^{\nu}_{\delta}+\bar{J}^{\beta}_{\gamma\delta}\right)$

where as before $J$ denotes an element of $GL(n)$ and $\bar{J}$ its inverse.
Another example is

$H(M)=\left(HM\doteq\mathrm{Lor}(M)\times_{M}(M\times\mathbb{R}^{2}),\pi,M,S^{2}_{(1,n-1)}(\mathbb{R}^{n})\times\mathbb{R}^{2}\right),$
(3.8)

where $S^{2}_{(1,n-1)}(\mathbb{R}^{m})$ denotes the space of signature
$(1,n-1)$ symmetric matrices. Coordinates will be given by fibered local
charts $(x^{\alpha},g_{\mu\nu},m,\kappa)$. We stress that since the
coefficients of the metric are symmetric then not all of them are independent,
therefore when we refer to $\\{g_{\mu\nu}\\}$ as coordinates we are really
considering just $\\{g_{\mu\nu}\\}_{1\leq\mu<\nu\leq n}$. Physical scaling on
$H(M)$ is then defined as follows: let $\lambda\in\mathbb{R}_{+}$, then we
define a fibered isomorphism projecting onto the identity of $M$ by

$\displaystyle S_{\lambda}:$ $\displaystyle HM\to HM,\quad$
$\displaystyle(x,g_{\mu\nu},m,\kappa)\mapsto(x,\lambda^{-2}g_{\mu\nu},\lambda^{2}m,\kappa).$
(3.9)

Here the coordinates $(m,\kappa)$ will later represent the mass and coupling
constant on the Klein-Gordon equation on $M$, for now however we take them to
just be coordinates of the fiber. We also note that $S_{\lambda}$ induces as
well a mapping $:\Gamma^{\infty}(M\leftarrow
HM)\rightarrow\Gamma^{\infty}(M\leftarrow HM):h\mapsto h_{\lambda}$ which we
denote by the same symbol.

######  Definition 3.1.6.

A natural bundle equipped with physical scaling is a natural functor

$H:\mathfrak{Man}\rightarrow\mathfrak{FBndl},$

from the category of manifolds and local diffeomorphisms to the category of
fiber bundles and local fiber isomorphisms, with the additional property that
for any $M\in Ob(\mathfrak{Man})$ there exists a corresponding physical
scaling $S_{\lambda}$ on $HM$ such that for any
$\psi\in\mathrm{Diff}_{loc}(M,M^{\prime})$ and $\lambda\in\mathbb{R}_{+}$ then

$S^{\prime}_{\lambda}\circ H(\psi)=H(\psi)\circ S_{\lambda}.$

Clearly (3.9) is an example of physical scaling according to Definition 3.1.6.

######  Definition 3.1.7.

Let $H:\mathfrak{Man}\rightarrow\mathfrak{Bndl}$ be the natural scaling bundle
described in (3.8), a background geometry is a pair $(M,h)$ where
$h\in\Gamma^{\infty}(M\leftarrow HM)$ and $(M,g)$ is a time orientable,
globally hyperbolic spacetime. Let then $\mathfrak{BckG}$ be the category of
background geometry, whose objects are time orientable globally hyperbolic
spacetimes and whose morphisms are time respecting, causality preserving
smooth isometric embeddings.

Note that in the above definition any such arrow $\chi:M\rightarrow
M^{\prime}$ of $\mathfrak{BckG}$, is in particular a local diffeomorphism, so
that $H(\chi)$ is defined according to Definition 3.1.1; moreover, the
transformation induced on the object $(M^{\prime},h^{\prime})$ is given by
$\left(M,h=\chi^{*}h^{\prime}\right)$, where the section $\chi^{*}h^{\prime}$
is defined as the mapping $H\left(\chi|_{H(\chi(M))}\right)^{-1}\circ
h^{\prime}\circ\chi$ guaranteeing the commutativity of the diagram

$HM$$HM^{\prime}$$M$$M^{\prime}$$H(\chi)$$\chi^{*}h^{\prime}$$h^{\prime}$$\chi$

We stress that the physical scaling introduced above is consistent with the
pullback of sections introduced above, due to commutativity with $H(\chi)$.

####  3.1.1 Covariance identity for natural bundles

In this section, we will present an important result (Theorem 3.1.9), which
states that if a smooth function $f$ from the $r$th order jet prolongation of
a natural bundle $J^{r}(\mathfrak{N}(M))$ to $\mathbb{R}$ is diffeomorphism
covariant, then it must satisfy some strict conditions: the coordinate
dependence is limited to all those coordinates of $J^{r}(\mathfrak{N}(M))$
which are tensor-like quantities, furthermore, $f$ can only be a function of
all the scalar quantities constructed out of the tensor-like coordinates
mentioned earlier. This result, which we will prove for the natural bundle
(3.8), will be crucial for the proof of Theorem 3.3.10, which essentially
classifies the ambiguities of the definition of Wick powers as scalars
constructed out of the matric and the coupling constants.

A useful consequence of the formalism of natural bundles is the possibility of
characterizing $\mathrm{Diff}(M)$-invariant functions on the $r$th jet order
$J^{r}\mathfrak{N}(M)$ for some natural bundle $\mathfrak{N}(M)$ by means of
the natural formalism developed in [26, 47]. Instead of analyzing the general
case we are going to study the case $\mathfrak{N}(M)=HM$ defined in (3.8).
There we assign coordinates $(x^{\mu},g_{\mu\nu},m^{2},\kappa)$, thus the
$r$th order jet prolongation has canonical coordinates

$(x^{\mu},g_{\mu\nu},\ldots,g_{\mu\nu,\lambda_{1}\ldots,\lambda_{r}},m^{2},\ldots,m^{2}_{\lambda_{1}\ldots,\lambda_{r}},\kappa,\ldots,\kappa_{\lambda_{1}\ldots,\lambda_{r}}).$

to derive the key result of covariance it is necessary to choose suitable
coordinates in $J^{r}HM$ other than the above.

######  Proposition 3.1.8.

Let $J^{r}HM$ be the natural bundle described above then, for each
$r\in\mathbb{R}$, there is a coordinates transformation

${(x^{\mu},g_{\mu\nu,\Lambda},m^{2}_{\Lambda},\kappa_{\Lambda})}$${(x^{\mu},g_{\mu\nu},d_{(\Lambda}S^{\alpha}_{\beta\mu)},\nabla_{(\Lambda}R^{\alpha}_{\beta\mu)\nu},\nabla_{(\Lambda)}m^{2},\nabla_{(\Lambda)}\kappa)}$

where $\Lambda=(\lambda_{1},\ldots,\lambda_{k})$ is a multi-index with
$k=|\Lambda|\leq r-2$, round brackets between indices denotes symmetrization,
the coefficients $S$ and $R$ are respectively the coefficients of the Levi-
Civita connection and the Riemann tensor induced by $g$; finally,
$d_{\mu}(\cdot)dx^{\mu}$ denotes the horizontal differential on $j^{r}HM$.

###### Proof.

We shall divide this argument into four steps: we shall analyse the cases
$r=1,2,3$ and then use those to formulate and induction hypothesis. We shall
also restrict to the simpler case $HM=\mathrm{Lor}(M)$ the bundle of
Lorentzian metrics on $M$ and come back to the general case at the end.

Step 1. We start with coordinates $(x^{\mu},g_{\mu\nu},d_{\alpha}g_{\mu\nu})$
on $J^{1}\mathrm{Lor}(M)$. Recall that the metric $g$ induces its Christoffel
symbols,
$S^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}(-d_{\beta}g_{\mu\nu}+d_{\mu}g_{\nu\beta}+d_{\nu}g_{\beta\mu})$,
then

$\displaystyle
2g_{\alpha\beta}S^{\alpha}_{\mu\nu}+2g_{\alpha\mu}S^{\alpha}_{\beta\nu}$
$\displaystyle={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{\mu}g_{\nu\beta}}+d_{\nu}g_{\beta\mu}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\beta}g_{\mu\nu}}$
$\displaystyle\quad+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\beta}g_{\nu\mu}}+d_{\nu}g_{\mu\beta}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{\mu}g_{\beta\nu}}$
$\displaystyle=2d_{\nu}g_{\mu\beta}.$

There are in total $n^{2}\frac{n+1}{2}$ independent components of
$d_{\nu}g_{\mu\beta}$ and of $S^{\alpha}_{\mu\nu}$, as a result

$\begin{cases}2d_{\nu}g_{\mu\beta}=2g_{\alpha\beta}S^{\alpha}_{\mu\nu}+2g_{\alpha\mu}S^{\alpha}_{\beta\nu}\\\
S^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}(-d_{\beta}g_{\mu\nu}+d_{\mu}g_{\nu\beta}+d_{\nu}g_{\beta\mu})\end{cases}$
(3.10)

are one the inverse to each other and produce a well defined change of
coordinates.

Step 2. We start with coordinates
$(x^{\mu},g_{\mu\nu},d_{\alpha}g_{\mu\nu},d_{\alpha\beta}g_{\mu\nu})$ on
$J^{2}\mathrm{Lor}(M)$, by (3.10), we can as well assume we have initial
coordinates
$(x^{\mu},g_{\mu\nu},S^{\alpha}_{\beta\nu},d_{\alpha\beta}g_{\mu\nu})$. Recall
that the Riemann tensor components are expressed as
$R^{\alpha}_{\beta\mu\nu}=d_{\mu}S^{\alpha}_{\beta\nu}+S^{\alpha}_{\sigma\mu}S^{\sigma}_{\beta\nu}-d_{\nu}S^{\alpha}_{\beta\mu}-S^{\alpha}_{\sigma\nu}S^{\sigma}_{\beta\mu}$.
Next define

$S^{\alpha}_{\beta\mu\nu}\doteq d_{(\mu}S^{\alpha}_{\beta\nu)}.$ (3.11)

we stress that round brackets means that all indices between them are
symmetrized.

$\displaystyle 3S^{\alpha}_{\beta\mu\nu}-2R^{\alpha}_{(\beta\nu)\mu}$
$\displaystyle=d_{\mu}S^{\alpha}_{\beta\nu}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{\beta}S^{\alpha}_{\nu\mu}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\nu}S^{\alpha}_{\mu\beta}}$
$\displaystyle\quad-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\nu}S^{\alpha}_{\beta\mu}}+d_{\mu}S^{\alpha}_{\beta\nu}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{\beta}S^{\alpha}_{\nu\mu}}+d_{\mu}S^{\alpha}_{\beta\nu}+O(j^{1}g)$
$\displaystyle=3d_{\mu}S^{\alpha}_{\beta\nu}+O(j^{1}g)$

where we denoted by $O(j^{1}g)$ in the above formula terms which depends at
most on $j^{1}g$ and for which (3.10) already gives the sought transformation.
Summing up we have

$d_{\mu}S^{\alpha}_{\beta\nu}=S^{\alpha}_{\beta\mu\nu}-\frac{2}{3}R^{\alpha}_{(\beta\nu)\mu}+\ldots$
(3.12)

where $\ldots$ refers to terms living in the first order jet bundle. Next we
compute

$\displaystyle d_{\alpha\beta}g_{\mu\nu}$
$\displaystyle=d_{\beta}\left(g_{\sigma\alpha}S^{\sigma}_{\mu\nu}+g_{\sigma\mu}S^{\sigma}_{\nu\alpha}\right)=g_{\sigma\alpha}d_{\beta}S^{\sigma}_{\mu\nu}+g_{\sigma\mu}d_{\beta}S^{\sigma}_{\nu\alpha}+O(j^{1}g)$
$\displaystyle=g_{\sigma\alpha}\left(S^{\sigma}_{\mu\beta\nu}-\frac{2}{3}R^{\sigma}_{(\mu\nu)\beta}\right)+g_{\sigma\mu}\left(S^{\sigma}_{\alpha\beta\nu}-\frac{2}{3}R^{\sigma}_{(\alpha\nu)\beta}\right)O(j^{1}g)$
$\displaystyle=2g_{\sigma(\alpha}S^{\sigma}_{\mu)\beta\nu}-\frac{1}{3}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}R_{\alpha\mu\nu\beta}}+R_{\alpha\nu\mu\beta}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}R_{\mu\alpha\nu\beta}}+R_{\mu\nu\alpha\beta}\right)+O(j^{1}g)$
$\displaystyle=2g_{\sigma(\alpha}S^{\sigma}_{\mu)\beta\nu}-\frac{1}{3}\left(R_{\alpha\nu\mu\beta}+R_{\alpha\beta\mu\nu}\right)+O(j^{1}g).$

Finally, $d_{\alpha\beta}g_{\mu\nu}$ has $n^{2}\left(\frac{n+1}{2}\right)^{2}$
independent components, while $dS$ and the Riemann tensor $R$ have
respectively $n\binom{n+2}{3}$ and $\frac{n^{2}}{12}(n+1)^{2}$ independent
components. It follows easily that
$n^{2}\left(\frac{n+1}{2}\right)^{2}=n\binom{n+2}{3}+\frac{n^{2}}{12}(n+1)^{2}$,
therefore we conclude that the mapping

$(x^{\mu},g_{\mu\nu},d_{\alpha}g_{\mu\nu},d_{\alpha\beta}g_{\mu\nu})\mapsto(x^{\mu},g_{\mu\nu},S^{\alpha}_{\mu\nu},S^{\alpha}_{\mu\beta\nu},R^{\alpha}_{\beta\mu\nu}),$

where up to order $1$ are defined by (3.13) and

$\begin{cases}d_{\alpha\beta}g_{\mu\nu}=2g_{\sigma(\alpha}S^{\sigma}_{\mu)\beta\nu}-\frac{1}{3}\left(R_{\alpha\nu\mu\beta}+R_{\alpha\beta\mu\nu}\right)+O(j^{1}g),\\\
S^{\alpha}_{\beta\mu\nu}\doteq d_{(\mu}S^{\alpha}_{\beta\nu)},\quad
R^{\alpha}_{\beta\mu\nu}=d_{\mu}S^{\alpha}_{\beta\nu}+S^{\alpha}_{\sigma\mu}S^{\sigma}_{\beta\nu}-d_{\nu}S^{\alpha}_{\beta\mu}-S^{\alpha}_{\sigma\nu}S^{\sigma}_{\beta\mu}\end{cases}$
(3.13)

is a well defined coordinate transformation.

Step 3. Then again, by (3.10), (3.13) we can assume to have initial
coordinates

$(x^{\mu},g_{\mu\nu},S^{\alpha}_{\mu\nu},S^{\alpha}_{\mu\beta\nu},R^{\alpha}_{\beta\mu\nu},d_{\alpha\beta\gamma}g_{\mu\nu})$

on $J^{3}\mathrm{Lor}(M)$. From (3.13),

$\displaystyle d_{\alpha\beta\gamma}g_{\mu\nu}$
$\displaystyle=2d_{\gamma}S_{(\alpha\mu)\beta\nu}-\frac{1}{3}\left(d_{\gamma}R_{\alpha\nu\mu\beta}+d_{\gamma}R_{\mu\nu\alpha\beta}\right)+O(j^{2}g)$
$\displaystyle=2d_{\gamma}S_{(\alpha\mu)\beta\nu}-\frac{1}{3}\left(\nabla_{\gamma}R_{\alpha\nu\mu\beta}+\nabla_{\gamma}R_{\mu\nu\alpha\beta}\right)+O(j^{2}g).$

Then define $S^{\alpha}_{\gamma\mu\beta\nu}\doteq
d_{(\gamma}S_{\alpha\mu\beta\nu)}$, which we can express as

$\displaystyle 4S^{\alpha}_{\gamma\mu\beta\nu}$
$\displaystyle=d_{\gamma}S^{\alpha}_{\mu\beta\nu}+d_{\mu}S^{\alpha}_{\beta\nu\gamma}+d_{\beta}S^{\alpha}_{\nu\gamma\mu}+d_{\nu}S^{\alpha}_{\gamma\mu\beta}$
$\displaystyle=d_{\gamma}S^{\alpha}_{\mu\beta\nu}+\frac{1}{3}\left(d_{\mu\nu}S^{\alpha}_{\beta\gamma}+d_{\mu\beta}S^{\alpha}_{\gamma\mu}+d_{\mu\gamma}S^{\alpha}_{\nu\beta}\right)$
$\displaystyle\quad+\frac{1}{3}\left(d_{\beta\gamma}S^{\alpha}_{\nu\mu}+d_{\beta\nu}S^{\alpha}_{\mu\gamma}+d_{\beta\mu}S^{\alpha}_{\gamma\nu}\right)$
$\displaystyle\quad+\frac{1}{3}\left(d_{\nu\mu}S^{\alpha}_{\beta\gamma}+d_{\nu\beta}S^{\alpha}_{\gamma\mu}+d_{\nu\gamma}S^{\alpha}_{\nu\beta}\right)$
$\displaystyle=d_{\gamma}S^{\alpha}_{\mu\beta\nu}+\frac{1}{3}d_{\gamma}\left(d_{\nu}S^{\alpha}_{\nu\beta}+d_{\mu}S^{\alpha}_{\nu\beta}+d_{\beta}S^{\alpha}_{\nu\mu}\right)$
$\displaystyle\quad+\frac{2}{3}\left(d_{\mu\nu}S^{\alpha}_{\beta\gamma}+d_{\mu\beta}S^{\alpha}_{\gamma\mu}+d_{\beta\nu}S^{\alpha}_{\mu\gamma}\right)$
$\displaystyle=2d_{\gamma}S^{\alpha}_{\mu\beta\nu}+\frac{2}{3}\left(d_{\mu\nu}S^{\alpha}_{\beta\gamma}+d_{\mu\beta}S^{\alpha}_{\nu\gamma}+d_{\beta\nu}S^{\alpha}_{\mu\gamma}\right).$

From the definition of the Riemann tensor we get
$d_{\gamma}R^{\alpha}_{\nu\mu\beta}=d_{\gamma\mu}S^{\alpha}_{\nu\beta}-d_{\gamma\beta}S^{\alpha}_{\nu\mu}+O(j^{2}g)$,
which plugged in the above expression yields

$\displaystyle 4S^{\alpha}_{\gamma\mu\beta\nu}$
$\displaystyle=2d_{\gamma}S^{\alpha}_{\mu\beta\nu}+\frac{2}{3}\left(d_{\mu\gamma}S^{\alpha}_{\beta\nu}-d_{\mu}R^{\alpha}_{\beta\gamma\nu}+d_{\beta\gamma}S^{\alpha}_{\nu\beta}-d_{\beta}R^{\alpha}_{\nu\gamma\mu}+d_{\gamma\nu}S^{\alpha}_{\mu\beta}+d_{\nu}R^{\alpha}_{\mu\beta\gamma}\right)+O(j^{2}g)$
$\displaystyle=4d_{\gamma}S^{\alpha}_{\mu\beta\nu}+\frac{2}{3}\left(d_{\mu}R^{\alpha}_{\beta\nu\gamma}+d_{\beta}R^{\alpha}_{\nu\mu\gamma}+d_{\nu}R^{\alpha}_{\mu\beta\gamma}\right)+O(j^{2}g)$
$\displaystyle=4d_{\gamma}S^{\alpha}_{\mu\beta\nu}+\frac{2}{3}\left(\nabla_{\mu}R^{\alpha}_{\beta\nu\gamma}+\nabla_{\beta}R^{\alpha}_{\nu\mu\gamma}+\nabla_{\nu}R^{\alpha}_{\mu\beta\gamma}\right)+O(j^{2}g),$

applying Bianchi identities simplifies the last term, and we get

$S^{\alpha}_{\gamma\mu\beta\nu}=d_{\gamma}S^{\alpha}_{\mu\beta\nu}+O(j^{2}g).$
(3.14)

Counting the independent components of $d_{\alpha\beta\gamma}g_{\mu\nu}$,
$S^{\alpha}_{\gamma\mu\beta\nu}$ and $\nabla_{\gamma}R^{\alpha}$ yields
respectively $\binom{n+1}{2}\binom{n+2}{3}$, $n\binom{n+3}{4}$ and
$\frac{n+2}{2}\frac{n^{2}}{12}(n+1)^{2}$. Summing the last two numebers gives
the first, therefore the mapping

$(x^{\mu},g_{\mu\nu},d_{\alpha}g_{\mu\nu},d_{\alpha\beta}g_{\mu\nu},d_{\alpha\beta\gamma}g_{\mu\nu})\mapsto(x^{\mu},g_{\mu\nu},S^{\alpha}_{\mu\nu},S^{\alpha}_{\mu\beta\nu},S^{\alpha}_{\gamma\mu\beta\nu},R^{\alpha}_{\beta\mu\nu},\nabla_{\gamma}R^{\alpha}_{\beta\mu\nu})$

is a proper change of coordinates, with

$\begin{cases}d_{\alpha\beta\gamma}g_{\mu\nu}=2g_{\sigma(\alpha}S^{\sigma}_{\beta)\gamma\mu\nu}-\frac{1}{3}\left(\nabla_{\gamma}R_{\alpha\nu\mu\beta}+\nabla_{\gamma}R_{\mu\nu\alpha\beta}\right)+O(j^{2}g).\\\
S^{\alpha}_{\gamma\mu\beta\nu}=d_{\gamma}S^{\alpha}_{\mu\beta\nu}+O(j^{2}g),\quad
d_{\gamma}R_{\alpha}{\nu\mu\beta}=\nabla_{\gamma}R_{\alpha}{\nu\mu\beta}+O(j^{2}g)\end{cases}$
(3.15)

Step 4.

By induction, we assume that up to $r-1\geq 3$, we have a one to one
coordinate transformation

$\begin{cases}d_{\alpha\beta\Lambda}g_{\mu\nu}=2g_{\sigma(\alpha}S^{\sigma}_{\beta)\Lambda\mu\nu}-\frac{1}{3}\left(\nabla_{\Lambda}R_{\alpha\nu\mu\beta}+\nabla_{\Lambda}R_{\mu\nu\alpha\beta}\right)+O(j^{r-2}g).\\\
d_{\beta}S^{\alpha}_{\Lambda\mu\nu}=S^{\alpha}_{\beta\Lambda\mu\nu}+O(j^{r-2}g),\quad
d_{\Lambda}R^{\alpha}_{\nu\mu\beta}=\nabla_{\Lambda}R^{\alpha}_{\nu\mu\beta}+O(j^{r-2}g)\end{cases}$
(3.16)

where $|\Lambda|=r-3$. Using the induction hypothesis
$d_{\beta}S^{\alpha}_{\Lambda\mu\nu}=S^{\alpha}_{\beta\Lambda\mu\nu}+O(j^{r-2}g)$
and repeating a similar argument to the one made for the case $r=3$, one can
show that

$S^{\alpha}_{\mu_{0}\ldots\mu_{r}}\doteq
d_{(\mu_{0}}S^{\alpha}_{\mu_{1}\ldots\mu_{r})}=d_{\mu_{0}}S^{\alpha}_{\mu_{1}\ldots\mu_{r}}+O(j^{r-1}g),$
(3.17)

Where similarly to what we did before, we denote by $O(j^{r-1}g)$ for a
polynomial function of the coordinates depending up to the $r-1$th order
derivatives of the metric. Then we have

$d_{\lambda_{1}\ldots\lambda{r}}g_{\mu\nu}=2g_{\sigma(\lambda_{1}}S^{\sigma}_{\mu)\nu\lambda_{2}\ldots\lambda_{r}}+\frac{2}{3}\nabla_{\lambda_{3}}\nabla_{\lambda_{4}\ldots\lambda_{r}}R_{\nu(\lambda_{1}\mu)\lambda_{2}}+O(j^{r-1}g),$

we can recast the above equation
$\nabla_{\lambda_{3}}\nabla_{\lambda_{4}\ldots\lambda_{r}}R_{\nu(\lambda_{1}\mu)\lambda_{2}}=\nabla_{\lambda_{3}\ldots\lambda_{r}}R_{\nu(\lambda_{1}\mu)\lambda_{2}}+O(j^{r-2}g)$
using the identity
$[\nabla_{\alpha},\nabla_{\beta}]=R^{\cdot}_{\cdot\alpha\beta}$ we get for
$|\Lambda|=r-2$,

$d_{\alpha\beta\Lambda}g_{\mu\nu}=2g_{\sigma(\alpha}S^{\sigma}_{\beta)\Lambda\mu\nu}-\frac{1}{3}\left(\nabla_{\Lambda}R_{\alpha\nu\mu\beta}+\nabla_{\Lambda}R_{\mu\nu\alpha\beta}\right)+O(j^{r-1}g)$
(3.18)

defines a proper coordinate transformation for the highest order derivatives
of the metric.

In our case where two scalar functions are present, we notice that each
symmetrized derivative, e.g. $d_{\Lambda}\kappa$, can be written as
$\nabla_{\Lambda}\kappa$, using the fact that
$\nabla_{\lambda}\kappa=d_{\lambda}\kappa$ and for all $\Lambda,\
\Lambda^{\prime}$, $|\Lambda|=k\geq 2$, $|\Lambda^{\prime}|=k-1$, we have
$\nabla_{\Lambda}\kappa-
d_{\Lambda}\kappa=O(\nabla_{\Lambda^{\prime}}\kappa,j^{|\Lambda|-1}g)$. ∎

######  Theorem 3.1.9.

If $f:J^{r}HM\rightarrow\mathbb{R}$ is a diffeomorphism covariant function,
then using the coordinates introduced in Proposition 3.1.8 it satisfies the
following relations:

  * $(i)$

$\frac{\partial f}{\partial x^{\sigma}}=0;$

  * $(ii)$

$\frac{\partial f}{\partial S^{\alpha}_{\mu\nu}}=\frac{\partial f}{\partial
S^{\alpha}_{\beta\mu\nu}}=\ldots=\frac{\partial f}{\partial
S^{\alpha}_{\lambda_{1}\ldots\lambda_{r}\mu}}=0;$

  * $(iii)$

furthermore, it holds

$\displaystyle 0$ $\displaystyle=2\frac{\partial f}{\partial
g_{\sigma\beta}}fg_{\rho\beta}+\frac{\partial f}{\partial
R^{\alpha}_{\beta\mu\nu}}\Big{(}-\delta^{\alpha}_{\rho}R^{\sigma}_{\beta\mu\nu}+\delta^{\sigma}_{\beta}R^{\alpha}_{\rho\mu\nu}+2\delta^{\sigma}_{\mu}R^{\alpha}_{\beta\rho\nu}\Big{)}+\ldots$
$\displaystyle\quad+\frac{\partial
f}{\partial\nabla_{\Lambda}R^{\alpha}_{\beta\mu\nu}}\Big{(}-\delta^{\alpha}_{\sigma}\nabla_{\Lambda}R^{\sigma}_{\beta\nu\mu}+\delta^{\sigma}_{\beta}\nabla_{\Lambda}R^{\alpha}_{\rho\nu\mu}+2\delta^{\sigma}_{\nu}\nabla_{\Lambda}R^{\alpha}_{\beta\rho\mu}+(r-3)\delta^{\sigma}_{\lambda_{1}}\nabla_{\rho\hat{\Lambda}}R^{\alpha}_{\beta\nu\mu}\Big{)}$
$\displaystyle\quad+\frac{\partial
f}{\partial\nabla_{\theta}m^{2}}\Big{(}\delta^{\sigma}_{\theta}\nabla_{\rho}m^{2}\Big{)}+\ldots+\frac{\partial
f}{\partial\nabla_{\Theta}m^{2}}\Big{(}-r\delta^{\sigma}_{\theta_{1}}\nabla_{\rho\hat{\Theta}}m^{2}\Big{)}$
$\displaystyle\quad+\frac{\partial
f}{\partial\nabla_{\theta}\kappa}\Big{(}\delta^{\sigma}_{\theta}\nabla_{\rho}\kappa\Big{)}+\ldots+\frac{\partial
f}{\partial\nabla_{\Theta}\kappa}\Big{(}-r\delta^{\sigma}_{\theta_{1}}\nabla_{\rho\hat{\Theta}}\kappa\Big{)}$

where $\Lambda=(\lambda_{1},\ldots,\lambda_{r-3})$ is a multi-index of length
$r-3$ and $\hat{\Lambda}=(\lambda_{2},\ldots,\lambda_{r-3})$.

Condition $(iii)$ essentially states that the function $f$ is a function whose
arguments are all covariant scalars that can be constructed out of the
coordinated of Proposition 3.1.8. These are always finite such objects.

###### Proof.

Then again, for simplicity’s sake, we study only the case where $f$ is a
smooth function in $J^{r}\mathrm{Lor}(M)$ and briefly comment at the end how
the adapt this argument in the general case.

Recall that since both the jet functor and the natural functor
$M\to\mathrm{Lor}(M)$ are covariant, then $N(M)=J^{r}\mathrm{Lor}(M)$ is a
natural bundle as well, thus, by Definition 3.1.1, we can lift any
$\chi\in\mathrm{Diff}(M)$ to an isomorphism $N\chi$ of $J^{r}\mathrm{Lor}(M)$.
$f$ is covariant whether $j^{r}\mathrm{Lor}(\chi)^{*}f=f$ for all
$\chi\in\mathrm{Diff}(M)$. If $\chi_{s}$ is a one parameter group of
diffeomorphisms of $M$, by $(iii)$ in Definition 3.1.1 we can lift it to a one
parameter isomorphism of $\mathrm{Lor}(M)$ and calling $X$, $\hat{X}$ the
respective infinitesimal generators, then

$j^{r}\mathrm{Lor}(\chi)^{*}f=f\Leftrightarrow\mathscr{L}_{j^{r}\hat{X}}f=0\Leftrightarrow
i_{X}d_{H}f=i_{j^{r}\mathscr{L}_{\hat{X}}g}d_{V}f,$ (3.19)

where $\mathscr{L}$ denotes the Lie derivative. In coordinates,
$\hat{X}=X^{\mu}\partial_{\mu}+X_{\mu\nu}\partial^{\mu\nu}$, we can then lift
it to a vector field in $\mathfrak{X}\left(J^{r}\mathrm{Lor}(M)\right)$ by

$\displaystyle\hat{X}$
$\displaystyle=X^{\mu}\partial_{\mu}+X_{\mu\nu}\partial^{\mu\nu}+X^{\beta}_{\mu\nu}\frac{\widetilde{\partial}}{\widetilde{\partial}S^{\beta}_{\mu\nu}}+\widetilde{X}^{\beta}_{\alpha\mu\nu}\frac{\widetilde{\partial}}{\widetilde{\partial}S^{\beta}_{\alpha\mu\nu}}+{X}^{\beta}_{\alpha\mu\nu}\frac{\partial}{\partial
R^{\beta}_{\alpha\mu\nu}}+\ldots$
$\displaystyle\quad+\widetilde{X}^{\beta}_{\lambda_{1}\ldots\lambda_{r}\mu}\frac{\widetilde{\partial}}{\widetilde{\partial}S^{\beta}_{\lambda_{1}\ldots\lambda_{r}\mu}}+{X}^{\beta}_{\lambda_{1}\ldots\lambda_{r}\mu}\frac{\partial}{\partial\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\beta}_{\lambda_{1}\mu\lambda_{2}}}.$

Calculations of the necessary Lie derivatives yields

$\displaystyle\mathscr{L}_{\hat{X}}g_{\alpha\beta}=2\nabla_{(\alpha}X^{\cdot}_{\beta)}$
$\displaystyle\mathscr{L}_{j^{1}\hat{X}}S^{\alpha}_{\mu\nu}=\nabla_{\mu\nu}X^{\alpha}-R^{\alpha}_{(\mu\nu)\rho}X^{\rho}$
$\displaystyle\mathscr{L}_{j^{2}\hat{X}}S^{\alpha}_{\beta\mu\nu}=\nabla_{\beta\mu\nu}X^{\alpha}+O(\nabla
X,X)$
$\displaystyle\mathscr{L}_{j^{2}\hat{X}}R^{\alpha}_{\beta\mu\nu}=\nabla_{\sigma}R^{\alpha}_{\beta\mu\nu}X^{\sigma}-\nabla_{\sigma}X^{\alpha}R^{\sigma}_{\beta\mu\nu}+\nabla_{\beta}X^{\sigma}R^{\alpha}_{\sigma\mu\nu}+\nabla_{\mu}X^{\sigma}R^{\alpha}_{\beta\sigma\nu}+\nabla_{\nu}X^{\sigma}R^{\alpha}_{\beta\mu\sigma}$
$\displaystyle\ldots$
$\displaystyle\mathscr{L}_{j^{r}\hat{X}}S^{\alpha}_{\lambda_{1}\ldots\lambda_{r}\mu}=\nabla_{\lambda_{1}\ldots\lambda_{r}\mu}X^{\alpha}+O(\nabla^{r-1}X,\ldots,X)$
$\displaystyle\mathscr{L}_{j^{r}\hat{X}}\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}=X^{\sigma}\nabla_{\sigma}\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}-\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\sigma}_{\lambda_{1}\mu\lambda_{2}}\nabla_{\sigma}X^{\alpha}+\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\sigma\mu\lambda_{2}}\nabla_{\lambda_{1}}X^{\sigma}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\ldots+\nabla_{\lambda_{3}\ldots\lambda_{r-1}\sigma}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}\nabla_{\lambda_{r}}X^{\sigma}+\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\sigma\lambda_{2}}\nabla_{\mu}X^{\sigma}$

Expanding (3.19) yields

$\displaystyle
X^{\sigma}\left(\partial_{\sigma}f+\partial^{\alpha\beta}f\partial_{\sigma}g_{\alpha\beta}+\partial^{\mu\nu}_{\alpha}f\partial_{\sigma}S^{\alpha}_{\mu\nu}+\widetilde{\partial}_{\alpha}^{\beta\mu\nu}f\partial_{\sigma}S^{\alpha}_{\beta\mu\nu}+{\partial}_{\alpha}^{\beta\mu\nu}f\partial_{\sigma}R^{\alpha}_{\beta\mu\nu}+\ldots\right.$
$\displaystyle\left.+\widetilde{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f\partial_{\sigma}S^{\alpha}_{\lambda_{1}\ldots\lambda_{r}\mu}+{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f\partial_{\sigma}\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}\right)=$
$\displaystyle
2\nabla_{(\alpha}X^{\cdot}_{\beta)}\partial^{\alpha\beta}f+\partial^{\mu\nu}_{\alpha}f\left(\nabla_{\mu\nu}X^{\alpha}+o(X)\right)+\widetilde{\partial}_{\alpha}^{\beta\mu\nu}f\left(\nabla_{\beta\mu\nu}X^{\alpha}+O(\nabla
X,X)\right)$
$\displaystyle+{\partial}_{\alpha}^{\beta\mu\nu}f\left(\nabla_{\sigma}R^{\alpha}_{\beta\mu\nu}X^{\sigma}-\nabla_{\sigma}X^{\alpha}R^{\sigma}_{\beta\mu\nu}+\nabla_{\beta}X^{\sigma}R^{\alpha}_{\sigma\mu\nu}+\nabla_{\mu}X^{\sigma}R^{\alpha}_{\beta\sigma\nu}+\nabla_{\nu}X^{\sigma}R^{\alpha}_{\beta\mu\sigma}\right)+$
$\displaystyle\widetilde{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f\left(\nabla_{\lambda_{1}\ldots\lambda_{r}\mu}X^{\alpha}+O(\nabla^{r-1}X,\ldots,X)\right)+{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f$
$\displaystyle\left(X^{\sigma}\nabla_{\sigma}\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}-\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\sigma}_{\lambda_{1}\mu\lambda_{2}}\nabla_{\sigma}X^{\alpha}+\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\sigma\mu\lambda_{2}}\nabla_{\lambda_{1}}X^{\sigma}\right.$
$\displaystyle\left.+\ldots+\nabla_{\lambda_{3}\ldots\lambda_{r-1}\sigma}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}\nabla_{\lambda_{r}}X^{\sigma}+\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\sigma\lambda_{2}}\nabla_{\mu}X^{\sigma}\right).$

Upon rewriting the partial derivative as covariant differential, we find

$\displaystyle
X^{\sigma}\left(\partial_{\sigma}f+2\partial^{\alpha\beta}fS^{\rho}_{\alpha\sigma}g_{\rho\beta}+\partial^{\mu\nu}_{\alpha}f\partial_{\sigma}S^{\alpha}_{\mu\nu}+\widetilde{\partial}_{\alpha}^{\beta\mu\nu}f\left(S^{\alpha}_{\sigma\beta\mu\nu}+O(j^{2}g)\right)\right.$
$\displaystyle\left.+{\partial}_{\alpha}^{\beta\mu\nu}f\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\nabla_{\sigma}R^{\alpha}_{\beta\mu\nu}}-S^{\alpha}_{\rho\sigma}R^{\rho}_{\beta\mu\nu}+S^{\rho}_{\beta\sigma}R^{\alpha}_{\rho\mu\nu}+2S^{\rho}_{\mu\sigma}R^{\alpha}_{\beta\rho\nu}\right)+\ldots\right.$
$\displaystyle\left.+\widetilde{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f\left(S^{\alpha}_{\sigma\lambda_{1}\ldots\lambda_{r}\mu}+O(j^{r})\right)+{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\nabla_{\sigma}\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}}-S^{\alpha}_{\rho\sigma}\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\sigma}_{\lambda_{1}\mu\lambda_{2}}\right.\right.$
$\displaystyle\left.\left.+S^{\rho}_{\lambda_{1}\sigma}\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\rho\mu\lambda_{2}}+2S^{\rho}_{\mu\sigma}\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\rho\lambda_{2}}+(r-3)S^{\rho}_{\lambda_{3}\sigma}\nabla_{\rho\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}\right)\right)=$
$\displaystyle
2\nabla_{(\alpha}X^{\cdot}_{\beta)}\partial^{\alpha\beta}f+\partial^{\mu\nu}_{\alpha}f\left(\nabla_{\mu\nu}X^{\alpha}+O(X)\right)+\widetilde{\partial}_{\alpha}^{\beta\mu\nu}f\left(\nabla_{\beta\mu\nu}X^{\alpha}+o(\nabla
X,X)\right)$
$\displaystyle+{\partial}_{\alpha}^{\beta\mu\nu}f\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\nabla_{\sigma}R^{\alpha}_{\beta\mu\nu}X^{\sigma}}-\nabla_{\sigma}X^{\alpha}R^{\sigma}_{\beta\mu\nu}+\nabla_{\beta}X^{\sigma}R^{\alpha}_{\sigma\mu\nu}+\nabla_{\mu}X^{\sigma}R^{\alpha}_{\beta\sigma\nu}+\nabla_{\nu}X^{\sigma}R^{\alpha}_{\beta\mu\sigma}\right)$
$\displaystyle+\widetilde{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f\left(\nabla_{\lambda_{1}\ldots\lambda_{r}\mu}X^{\alpha}+O(\nabla^{r-1}X,\ldots,X)\right)+{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f$
$\displaystyle\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}X^{\sigma}\nabla_{\sigma}\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}}-\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\sigma}_{\lambda_{1}\mu\lambda_{2}}\nabla_{\sigma}X^{\alpha}+\nabla_{\lambda_{3}\ldots,\lambda_{r}}R^{\alpha}_{\sigma\mu\lambda_{2}}\nabla_{\lambda_{1}}X^{\sigma}\right.$
$\displaystyle\left.+\ldots+\nabla_{\lambda_{3}\ldots\lambda_{r-1}\sigma}R^{\alpha}_{\lambda_{1}\mu\lambda_{2}}\nabla_{\lambda_{r}}X^{\sigma}+\nabla_{\lambda_{3}\ldots\lambda_{r}}R^{\alpha}_{\lambda_{1}\sigma\lambda_{2}}\nabla_{\mu}X^{\sigma}\right).$

Since the vector field $X\in\mathfrak{X}(M)$ is arbitrary, the above equation
must hold for any order of covariant derivative in $X$ independently. As a
result, we recursively find that
$\widetilde{\partial}_{\alpha}^{\lambda_{1}\ldots\lambda_{r}\mu}f=\ldots=\widetilde{\partial}_{\alpha}^{\beta\mu\nu}f=\partial^{\mu\nu}_{\alpha}f=0$.
Then collecting the $X^{\sigma}$ and $\nabla_{\sigma}X^{\rho}$ terms we get
the system

$\begin{cases}\begin{aligned}
&0=\partial_{\sigma}f+S^{\rho}_{\gamma\sigma}\Big{\\{}2\partial^{\sigma\beta}fg_{\rho\beta}+{\partial}_{\alpha}^{\beta\mu\nu}f\left(-\delta^{\alpha}_{\rho}R^{\sigma}_{\beta\mu\nu}+\delta^{\sigma}_{\beta}R^{\alpha}_{\rho\mu\nu}+2\delta^{\sigma}_{\mu}R^{\alpha}_{\beta\rho\nu}\right)+\ldots\\\
&+{\partial}_{\alpha}^{\beta\mu\Lambda\nu}f\left(-\delta^{\alpha}_{\sigma}\nabla_{\Lambda}R^{\sigma}_{\beta\nu\mu}+\delta^{\sigma}_{\beta}\nabla_{\Lambda}R^{\alpha}_{\rho\nu\mu}+2\delta^{\sigma}_{\nu}\nabla_{\Lambda}R^{\alpha}_{\beta\rho\mu}+(r-3)\delta^{\sigma}_{\lambda_{1}}\nabla_{\rho\hat{\Lambda}}R^{\alpha}_{\beta\nu\mu}\right)\Big{\\}},\end{aligned}\\\
\begin{aligned}
&0=2\partial^{\sigma\beta}fg_{\rho\beta}+{\partial}_{\alpha}^{\beta\mu\nu}f\left(-\delta^{\alpha}_{\rho}R^{\sigma}_{\beta\mu\nu}+\delta^{\sigma}_{\beta}R^{\alpha}_{\rho\mu\nu}+2\delta^{\sigma}_{\mu}R^{\alpha}_{\beta\rho\nu}\right)+\ldots\\\
&+{\partial}_{\alpha}^{\beta\mu\Lambda\nu}f\left(-\delta^{\alpha}_{\sigma}\nabla_{\Lambda}R^{\sigma}_{\beta\nu\mu}+\delta^{\sigma}_{\beta}\nabla_{\Lambda}R^{\alpha}_{\rho\nu\mu}+2\delta^{\sigma}_{\nu}\nabla_{\Lambda}R^{\alpha}_{\beta\rho\mu}+(r-3)\delta^{\sigma}_{\lambda_{1}}\nabla_{\rho\hat{\Lambda}}R^{\alpha}_{\beta\nu\mu}\right).\end{aligned}\end{cases}$

substituting the second equation into the first we get the initial claim

$\displaystyle 0$ $\displaystyle=2\frac{\partial f}{\partial
g_{\sigma\beta}}fg_{\rho\beta}+\frac{\partial f}{\partial
R^{\alpha}_{\beta\mu\nu}}\Big{(}-\delta^{\alpha}_{\rho}R^{\sigma}_{\beta\mu\nu}+\delta^{\sigma}_{\beta}R^{\alpha}_{\rho\mu\nu}+2\delta^{\sigma}_{\mu}R^{\alpha}_{\beta\rho\nu}\Big{)}+\ldots$
$\displaystyle\quad+\frac{\partial
f}{\partial\nabla_{\Lambda}R^{\alpha}_{\beta\mu\nu}}\Big{(}-\delta^{\alpha}_{\sigma}\nabla_{\Lambda}R^{\sigma}_{\beta\nu\mu}+\delta^{\sigma}_{\beta}\nabla_{\Lambda}R^{\alpha}_{\rho\nu\mu}+2\delta^{\sigma}_{\nu}\nabla_{\Lambda}R^{\alpha}_{\beta\rho\mu}+(r-3)\delta^{\sigma}_{\lambda_{1}}\nabla_{\rho\hat{\Lambda}}R^{\alpha}_{\beta\nu\mu}\Big{)}.$

To include the scalars $m^{2},\ \kappa$ we simply note that being scalar
coordinates, their covariant derivatives are tensors, therefore diff-covariant
functions are allowed to depend on them, and the above equation gets modified
by adding to the right hand side the term

$\displaystyle\frac{\partial
f}{\partial\nabla_{\theta}m^{2}}\Big{(}\delta^{\sigma}_{\theta}\nabla_{\rho}m^{2}\Big{)}+\ldots+\frac{\partial
f}{\partial\nabla_{\Theta}m^{2}}\Big{(}-r\delta^{\sigma}_{\theta_{1}}\nabla_{\rho\hat{\Theta}}m^{2}\Big{)}+\frac{\partial
f}{\partial\nabla_{\theta}\kappa}\Big{(}\delta^{\sigma}_{\theta}\nabla_{\rho}\kappa\Big{)}+\ldots+\frac{\partial
f}{\partial\nabla_{\Theta}\kappa}\Big{(}-r\delta^{\sigma}_{\theta_{1}}\nabla_{\rho\hat{\Theta}}\kappa\Big{)}$

with $|\Theta|=r$. ∎

###  3.2 Hadamard parametrices

In this section we will review some of the geometry needed to write locally
the Hadamard parametrix, the latter is the closest thing (from a
microanalytical standpoint) to a solution of some partial differential
equation: the parametrix is a distribution which is smoothened by the
differential operator inducing the linear equations. In particular we are
interested in wave equations such as (2.84). The construction that we are
describing can be readily generalized from the scalar to the vector case (e.g.
see [29, $\S$ 6], [15, $\S$ A] for the Euclidean case).

Let $(M,g)$ be a time oriented Lorentzian manifold, given some point $x\in M$
consider the pseudo-Riemannian exponential mapping $\mathrm{exp}_{x}:V\subset
T_{x}M\to M$ of the metric $g$. Since $T_{0}\mathrm{exp}_{x}=id_{T_{x}M}$,
there will be a neighborhood $\Omega\ni x$ such that, possibly restricting the
domain, $\mathrm{exp}_{x}:V\subset T_{x}M\to\Omega$ is a diffeomorphism.
Alternatively, we can say that there are neighborhoods $\mathcal{O}\subset
M^{2}$ of the diagonal $\Delta_{2}(M)$, $\widetilde{V}\subset TM$ of the graph
of the zero section, such that $\mathrm{exp}:\widetilde{V}\subset
TM\to\mathcal{O}$ is a diffeomorphism. The square geodesic distance function
is then defined as

$\sigma:\mathcal{O}\to\mathbb{R}:(x,y)\mapsto\sigma(x,y)=\frac{1}{2}g(x)\big{(}\mathrm{exp}_{x}^{-1}(y),\mathrm{exp}_{x}^{-1}(y)\big{)}$
(3.20)

In this we shall adopt a particular notation: let $(x,y)\in\mathcal{O}$,
$v=\mathrm{exp}^{-1}_{x}(y),\ w=\mathrm{exp}_{y}^{-1}(x)$, we denote by
$(U_{x},x^{\mu})$, $(U_{y},x^{\alpha})$ the coordinates around $x$, $y$
respectively, and by $(V,\dot{x}^{\mu})$, $(V,\dot{x}^{\alpha})$ the
coordinates of $T_{x}M$, $T_{y}M$. In short, Greek indices such as
$\alpha,\beta,\gamma,\delta$ will denote coordinates around $y$ of $T_{y}M$,
indices such as $\mu,\nu,\lambda,\rho$ will refer to coordinates around $x$ or
$T_{x}M$. In those coordinates, set
$(\mathrm{exp}_{x}(v))^{\alpha}=e^{\alpha}_{x}(v)$,
$(\mathrm{exp}^{-1}_{x}(y))^{\mu}=\bar{e}^{\mu}_{x}(y)$,
$(\mathrm{exp}_{y}(w))^{\mu}=e^{\mu}_{y}(w)$,
$(\mathrm{exp}^{-1}_{y}(x))^{\alpha}=\bar{e}^{\alpha}_{y}(x)$. Since
$\mathrm{exp}_{x}(\mathrm{exp}_{x}^{-1}(y))=y$,
$\mathrm{exp}_{x}^{-1}(\mathrm{exp}_{x}(v))=v$, applying the tangent mapping
to both sides of those equations yield the identities

$T_{\mathrm{exp}^{-1}_{x}(y)}\mathrm{exp}_{x}\circ
T_{y}\mathrm{exp}_{x}^{-1}=T_{y}id\ \Rightarrow\frac{\partial
e_{x}^{\alpha}}{\partial\dot{x}^{\mu}}\frac{\partial\bar{e}_{x}^{\mu}}{\partial
y^{\beta}}=\delta^{\alpha}_{\beta};\ \frac{\partial
e_{y}^{\mu}}{\partial\dot{y}^{\alpha}}\frac{\partial\bar{e}_{x}^{\alpha}}{\partial
y^{\nu}}=\delta^{\mu}_{\nu},$

$T_{\mathrm{exp}_{x}(v)}\mathrm{exp}^{-1}_{x}\circ
T_{v}\mathrm{exp}_{x}=T_{v}id,\
\Rightarrow\frac{\partial\bar{e}_{x}^{\mu}}{\partial y^{\alpha}}\frac{\partial
e_{x}^{\alpha}}{\partial\dot{x}^{\nu}}=\delta^{\mu}_{\nu},\
\frac{\partial\bar{e}_{y}^{\alpha}}{\partial x^{\mu}}\frac{\partial
e_{y}^{\mu}}{\partial\dot{y}^{\beta}}=\delta^{\alpha}_{\beta};$

######  Lemma 3.2.1.

The square geodesic distance (3.20) satisfies the following identity:

$\sigma=\frac{1}{2}g^{\mu\nu}(x)\nabla_{\mu}\sigma\nabla_{\nu}\sigma=\frac{1}{2}g^{\alpha\beta}(y)\nabla_{\alpha}\sigma\nabla_{\beta}\sigma.$
(3.21)

Moreover, as $y\to x$, we have the identities

$\displaystyle\sigma_{\alpha}\to 0;$ (3.22)
$\displaystyle\sigma_{\alpha\beta}\to g_{\alpha\beta}(x);$ (3.23)
$\displaystyle\sigma_{\alpha\beta\gamma}\to 0;$ (3.24)
$\displaystyle\sigma_{\alpha\beta\gamma\delta}\to\frac{1}{3}\big{(}R_{\alpha\gamma\beta\delta}(x)+R_{\alpha\delta\beta\gamma}(x)\big{)};$
(3.25)

where $\sigma_{\alpha}=\nabla_{\alpha}\sigma$, and $R$ the Riemann tensor of
the metric $g$. Finally, if $y\in I^{+}(x)$ $($resp. $y\in I^{-}(x)$ $)$ then
$\sigma_{\alpha}$ is past $($resp. future$)$ directed.

###### Proof.

First recall that Gauss lemma, which states that $\mathrm{exp}_{x}$ is a
radial isometry, that is,

$g(y)\big{(}T_{v}\mathrm{exp}_{x}(v),T_{v}\mathrm{exp}_{x}(v^{\prime})\big{)}=g(x)(v,v^{\prime}),\
\forall v,v^{\prime}\in T_{x}M,\ y=\mathrm{exp}_{x}(v);$

or, in coordinate notation,

$g_{\alpha\beta}(y)\frac{\partial
e_{x}^{\alpha}}{\partial\dot{x}^{\mu}}\frac{\partial
e_{x}^{\beta}}{\partial\dot{x}^{\nu}}=g_{\mu\nu}(x),\
g^{\alpha\beta}(y)\frac{\partial\bar{e}_{x}^{\mu}}{\partial{y}^{\alpha}}\frac{\partial\bar{e}_{x}^{\nu}}{\partial{y}^{\beta}}=g^{\mu\nu}(x)$
(3.26)

Then,

$\displaystyle\frac{1}{2}g^{\alpha\beta}(y)\nabla_{\alpha}\sigma\nabla_{\beta}\sigma$
$\displaystyle=\frac{1}{8}g^{\alpha\beta}(y)\nabla_{\alpha}\big{(}g_{\mu\nu}(x)\bar{e}^{\mu}_{x}(y)\bar{e}^{\nu}_{x}(y)\big{)}\big{(}g_{\lambda\rho}(x)\bar{e}^{\lambda}_{x}(y)\bar{e}^{\rho}_{x}(y)\big{)}$
$\displaystyle=\frac{1}{2}g^{\alpha\beta}(y)g_{\mu\nu}(x)\frac{\bar{e}^{\mu}_{x}(y)}{\partial
y^{\alpha}}\bar{e}^{\nu}_{x}(y)g_{\lambda\rho}(x)\frac{\bar{e}^{\lambda}_{x}(y)}{\partial
y^{\beta}}\bar{e}^{\rho}_{x}(y)$
$\displaystyle=\frac{1}{2}g^{\mu\lambda}(x)g_{\mu\nu}(x)g_{\lambda\rho}(x)e^{\nu}_{x}(y)\bar{e}^{\rho}_{x}(y)=\sigma(x,y).$

where in the third equality we used (3.26). The identities (3.22) follows from
$\sigma\to 0$ as $y\to x$ and repeated differentiation of (3.21). For details
see [20, pp. 227-228]. Finally, suppose $y\in I^{+}(x)$, then $\sigma(x,y)<0$,
by (3.21), we must have that $\sigma_{\alpha}$ is timelike. Let $T$ be the
vector field inducing the time orientation on $M$ and $\gamma_{x}(t)$ the
integral curve of $T$ starting at $x$ with velocity $T(x)\in T_{x}M$, by
construction the function $t\mapsto\sigma(x,\gamma_{x}(t))$ is negative and
decreasing, thus

$0>\frac{d}{dt}\sigma(x,\gamma_{x}(t))=g^{\alpha\beta}(y)\big{(}\sigma_{\alpha}|_{\gamma_{x}(t)},T(\gamma_{x}(t))\big{)}.$

∎

The geodesic parallel displacement is a mapping

$\bar{\delta}:\mathcal{O}\to\Omega_{1}(M)\boxtimes\Omega_{1}(M):(x,y)\mapsto\bar{\delta}_{\alpha\mu}dy^{\alpha}\boxtimes
dx^{\mu},$ (3.27)

satisfying the following requirements:

$\lim_{y\to
x}g^{\alpha\beta}(y)\bar{\delta}_{\beta\mu}(x,y)=\delta^{\alpha}_{\mu},$
(3.28) $g^{\mu\nu}(x)\sigma_{\mu}\nabla_{\nu}\bar{\delta}_{\alpha\lambda}=0,\
g^{\alpha\beta}(y)\sigma_{\alpha}\nabla_{\beta}\bar{\delta}_{\gamma\mu}=0.$
(3.29)

We stress that $\bar{\delta}$ is the mapping condition (3.29) is the equation
of parallel transport, from $T_{x}M$ to $T_{y}M$, along the geodesic joining
$x$ to $y$ whereas (3.28) is the initial condition. From those condition the
existence and uniqueness of the geodesic parallel displacement can be derived.

######  Lemma 3.2.2.

The geodesic parallel displacement introduced in (3.27) satisfies the
following relations:

  * $(i)$

$\bar{\delta}^{\alpha}_{\ \mu}\sigma_{\alpha}=-\sigma_{\mu}$,
$\bar{\delta}_{\alpha}^{\ \mu}\sigma_{\mu}=-\sigma_{\alpha}$;

  * $(ii)$

$\bar{\delta}^{\alpha\mu}\bar{\delta}_{\alpha\nu}=\delta^{\mu}_{\nu}$,
$\bar{\delta}^{\alpha\mu}\bar{\delta}_{\beta\mu}=\delta^{\alpha}_{\beta}$;

  * $(iii)$

$\bar{\delta}^{\alpha}_{\ \mu}\bar{\delta}^{\beta}_{\
\nu}g_{\alpha\beta}(y)=g_{\mu\nu}(x)$, $\bar{\delta}^{\ \mu}_{\
\alpha}\bar{\delta}_{\beta}^{\ \nu}g_{\mu\nu}(x)=g_{\alpha\beta}(y)$.

###### Proof.

$(i)$ follows from the fact that if $y=\exp_{x}(v)$, the geodesic joining the
two points is $\gamma(t)=\exp_{x}(tv)$, calling $w=-\exp^{-1}_{x}(y)$, we have
$\gamma(1-t)=\exp_{y}(tw)$, then we have

$\frac{\partial
e_{x}^{\alpha}}{\partial\dot{x}^{\mu}}\bar{e}_{x}^{\mu}=-\bar{e}_{y}^{\alpha}.$
(3.30)

Using the above equality and (3.26) we get
$g_{\mu\nu}(x)\sigma_{\nu}=-\bar{e}_{y}^{\alpha}$. Then

$\bar{\delta}_{\alpha}^{\ \mu}\sigma_{\mu}=\frac{\partial
e_{x}^{\alpha}}{\partial\dot{x}^{\mu}}\bar{e}_{x}^{\mu}=-\bar{e}_{y}^{\alpha}=-\sigma_{\alpha}.$

$(ii)$ states that the composition of parallel transport from $x$ to $y$ and
back is the identity, and can be seen using (3.29) to show that for any $v\in
T_{x}M$,

$\sigma^{\lambda}\nabla_{\lambda}\big{(}\bar{\delta}^{\alpha\mu}\bar{\delta}_{\alpha\nu}v^{\nu}\big{)}=0.$

Finally, $(iii)$ follows from the fact that along the geodesic
$\gamma(t)=e_{x}(t\bar{e}_{x}(y))$, each $v,v^{\prime}\in T_{x}M$ is
transported parallel to the geodesic and thus the scalar product of the two
vector remain constant along $\gamma$. Therefore expanding

$0=\frac{d}{dt}\Big{(}g(\gamma(t))\big{(}\delta(v),\delta(v^{\prime})\big{)}\Big{)}$

in coordinates we conclude. ∎

We now introduce three quantities that are of great importance to define the
Hadamard parametrix:

$\bar{\delta}(x,y)=\mathrm{det}(\bar{\delta}^{\alpha}_{\ \mu}),$ (3.31)

Notice that by $(i)$ in Lemma 3.2.2, we have
$g^{\mu\nu}\bar{\delta}^{\alpha}_{\ \mu}\sigma_{\alpha}\bar{\delta}^{\beta}_{\
\nu}\sigma_{\beta}=g^{\mu\nu}\sigma_{\mu}\sigma_{\nu}=\sigma$, implying
$g_{\mu\nu}=\bar{\delta}^{\alpha}_{\
\mu}g_{\alpha\beta}\bar{\delta}^{\beta}_{\ \nu}$. Taking the determinant we
get

$\bar{\delta}(x,y)=g^{1/2}(x)g^{1/2}(y).$ (3.32)

The second quantity of interest is van Vleck determinant:

$D(x,y)=-\mathrm{det}(\sigma_{\alpha\mu}),$ (3.33)

it’s easy to show, arguing with (3.26) and (3.30), that

$-\sigma_{\mu\alpha}(x,y)=g_{\alpha\beta}(y)\frac{\partial\bar{e}_{y}^{\beta}}{\partial
x^{\mu}}.$ (3.34)

Finally, define

$\Delta(x,y)=\frac{D(x,y)}{\bar{\delta}(x,y)}.$ (3.35)

######  Lemma 3.2.3.

The quantity $\Delta$ defined in (3.35) satisfies

$g^{\alpha\beta}\nabla_{\alpha}\big{(}\Delta\sigma_{\beta}\big{)}=n\Delta,$

where $n$ is the dimension of $M$.

###### Proof.

By explicit calculation,

$\displaystyle g^{\alpha\beta}\nabla_{\alpha}\big{(}\Delta\sigma_{\nu}\big{)}$
$\displaystyle=g^{\alpha\beta}\nabla_{\alpha}\Delta\sigma_{\beta}+g^{\alpha\beta}\Delta\sigma_{\alpha\beta}$
$\displaystyle=-\Delta
g^{\alpha\beta}\delta_{\epsilon\mu}\nabla_{\gamma}\bar{\delta}^{\epsilon\mu}\sigma_{\beta}+\Delta
g^{\alpha\beta}\sigma_{\epsilon\mu}\nabla_{\gamma}\sigma^{\epsilon\mu}\sigma_{\beta}+g^{\alpha\beta}\Delta\sigma_{\alpha\beta},$

the first term is zero by (3.29), where as the rest can be written as

$\Delta
g^{\alpha\beta}\sigma^{\gamma\mu}(\nabla_{\alpha}\sigma_{\gamma\mu}\sigma_{\beta}+\sigma_{\alpha\gamma}\sigma_{\mu\beta}),$

Notice that,

$\sigma_{\alpha\mu}=\nabla_{\alpha}\sigma_{\mu}=\nabla_{\alpha}(g^{\rho\nu}\sigma_{\mu\nu}\sigma_{\rho})=g^{\rho\nu}(\nabla_{\alpha}\sigma_{\mu\nu}+\sigma_{\alpha\rho}),$

which, using (3.34) to calculate $\sigma^{\gamma\mu}\sigma_{\gamma\mu}$,
implies

$g^{\alpha\beta}\nabla_{\alpha}\big{(}\Delta\sigma_{\nu}\big{)}=\Delta\sigma^{\gamma\mu}\sigma_{\gamma\mu}=n\Delta.$

∎

We are going to use this identities to construct solutions for wave equations
of the form $P=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}+B$, where $B\in
C^{\infty}(M)$. i.e. equations defined by normally hyperbolic differential
operators c.f. Lemma 1.2.12. We are, in particular, looking for a parametrix
for the equations, i.e. a distribution
$H\in\mathcal{D}^{\prime}(\mathcal{O})$, such that if $x\in M$ is fixed,
$\Omega\ni x$ is a geodesically convex neighborhood,

$P(y)H(x,y)-\delta_{y}\in C^{\infty}(\Omega).$ (3.36)

In the above expression $\delta_{y}:\mathcal{E}(\Omega)\ni f\mapsto
f(y)\in\mathbb{R}$ is Dirac’s delta. Denote by $\mathrm{Par}(M,h)$ the set of
parametrices with respect to the normally hyperbolic operator $P$.

######  Theorem 3.2.4.

Suppose that $(M,g)$ is a time oriented Lorentzian manifold let
$d=\frac{n-2}{2}$, then given any wave equation of the form $P=\square_{g}+B$
with $B\in C^{\infty}(M)$, there exist local parametrices:

  * $(i)$

if $n$ is even,

$H(x,y)=\beta_{n}\bigg{[}\frac{U(x,y)}{(\sigma(x,y)+i\epsilon
t(x,y)+\epsilon^{2})^{d}}+V(x,y)\ln\Big{(}\frac{1}{\mu^{2}}(\sigma+i\epsilon
t(x,y)+\epsilon^{2})\Big{)}+W(x,y)\bigg{]},$

  * $(ii)$

if $n$ is odd,

$H(x,y)=\beta_{n}\frac{Z(x,y)}{(\sigma(x,y)+i\epsilon
t(x,y)+\epsilon^{2})^{d}},$

where $\mu$ is a constant length scale, $t(x,y)\equiv t(x)-t(y)$ and $t$ is
the Cauchy temporal function, $\epsilon\to 0$ the parameter used to evaluate
the Hadamard principal value of the integral and the cut in the complex domain
of $\ln$ is in the negative real axis. The coefficients
$U=\sum_{j=1}^{d-1}U_{j}\sigma^{j}$, $V=\sum_{j\geq
0}V_{j}\sigma^{j}\rho(\sigma k_{j})$, $W=\sum_{j\geq
0}W_{j}\sigma^{j}\rho(\sigma k_{j})$, $Z=\sum_{j\geq
0}Z_{j}\sigma^{j}\rho(\sigma k_{j})$ are recursively constructed out of the
geometric data of the problem, $\rho\in C^{\infty}_{c}(\mathbb{R})$ satisfies
$0\leq\rho\leq 1$, $\rho(z)\equiv 1$ for $|z|\leq 1/2$, $\rho(z)\equiv 0$ for
$|z|\geq 1$ and $\\{k_{j}\\}_{j\in\mathbb{N}}$ is a suitable sequence with
$k_{j}\nearrow\infty$. Each function $U_{j},V_{j},W_{j},Z_{j}\in
C^{\infty}(\Omega\times\Omega)$ is inductively constructed out of the metric
$g$ and the operator $P$ and $\beta_{n}$ are numerical coefficients depending
on the dimension, moreover each series is convergent to a smooth function of
$C^{\infty}(\Omega\times\Omega)$ for a suitable geodesically convex
neighborhood $\Omega$ of $x\in M$. Moreover, given $H_{1},\
H_{2}\in\mathrm{Par}(M,h)$, $H_{1}-H_{2}\in C^{\infty}(\mathcal{O})$.

###### Proof.

For simplicity we will do the case $n=4$, however, similar arguments also
apply to the other cases; for the specific details we point to [31]. For
simplicity, we also set $\mu=1$, the general case being a straightforward
generalization. We divide the proof in steps: first we define the coefficients
$U,\ V,\ W$ by requiring that $PH=0$ at each order in $\sigma$, then we take
care of the convergence of the newly determined infinite series by introducing
some ad-hoc cutoff factor which ensure convergence in the Fréchet topology of
$C^{\infty}(\mathcal{O})$ but weakens $PH=0$ into $PH\in
C^{\infty}(\mathcal{O})$, finally, using the results in [44, Appendix B] we
show that $H$ is a well defined distribution.

Step 1. Fixed $x\in M$ consider

$\displaystyle P(y)H(x,y)$
$\displaystyle=-\sigma^{-2}g^{\alpha\beta}\big{(}2\nabla_{\alpha}U-U\Delta^{-1}\nabla_{\alpha}\Delta\big{)}\sigma_{\beta}$
$\displaystyle\quad+\sigma^{-1}\big{[}2V+g^{\alpha\beta}\big{(}2\nabla_{\alpha}V-V\Delta^{-1}\nabla_{\alpha}\Delta\big{)}\sigma_{\beta}+PU\big{]}$
$\displaystyle\quad+PV\ln(\sigma)+PW.$

setting

$U(x,y)=\Delta^{1/2}(x,y)$ (3.37)

and using Lemma 3.2.3 we get
$U^{-1}\nabla_{\alpha}U=\frac{1}{2}\Delta^{-1}\nabla_{\alpha}\Delta$, which
substituted into the above expression for $P(y)H(x,y)$ yields, at each order
in $\sigma$, the following identities

$\displaystyle
V_{0}+g^{\alpha\beta}\big{(}\nabla_{\alpha}V_{0}-\frac{1}{2}V_{0}\Delta^{-1}\nabla_{\alpha}\Delta)\sigma_{\beta}$
$\displaystyle=-\frac{1}{2}PU$
$\displaystyle(j+1)V_{j}+g^{\alpha\beta}\big{(}\nabla_{\alpha}V_{j}-\frac{1}{2}V_{j}\Delta^{-1}\nabla_{\alpha}\Delta\big{)}\sigma_{\beta}$
$\displaystyle=-\frac{1}{2j}PV_{j-1}$
$\displaystyle(j+1)W_{j}+g^{\alpha\beta}\big{(}\nabla_{\alpha}W_{j}-\frac{1}{2}W_{j}\Delta^{-1}\nabla_{\alpha}\Delta\big{)}\sigma_{\beta}$
$\displaystyle=-\frac{1}{2j}P(W_{j-1}-V_{j-1})-V_{n}$

To solve the above system we use normal coordinates
$\dot{x}^{\mu}=\bar{e}_{x}^{\mu}(y)$ around $x$, in those coordinates

$U(x,\dot{x})=\Big{(}\frac{g(x)}{g(\dot{x})}\Big{)}^{1/4}$ (3.38)

then we get the system

$\displaystyle\dot{x}^{\mu}\partial_{\mu}\Big{(}\frac{V_{0}}{U}\Big{)}+\frac{V_{0}}{U}$
$\displaystyle=-\frac{1}{4}\frac{PU}{U},$
$\displaystyle\dot{x}^{\mu}\partial_{\mu}\Big{(}\frac{V_{j}}{U}\Big{)}+(j+1)\frac{V_{j}}{U}$
$\displaystyle=-\frac{1}{4}\frac{PV_{j-1}}{U},$
$\displaystyle\dot{x}^{\mu}\partial_{\mu}\Big{(}\frac{W_{j}}{U}\Big{)}+(j+1)\frac{W_{j}}{U}$
$\displaystyle=-\frac{1}{4}\frac{PW_{j-1}-PV_{j-1}}{U}-\frac{V_{j}}{U},$

which can be integrated as

$V_{0}(x,\dot{x})=-\frac{U(x,\dot{x})}{4}\int_{0}^{1}\frac{PU(x,s\dot{x}^{\mu})}{U(x,s\dot{x}^{\mu})}ds.$
(3.39)
$V_{l}(x,\dot{x})=-\frac{U(x,\dot{x})}{4l}\int_{0}^{1}\frac{s^{l}PV_{l-1}(x,s\dot{x}^{\mu})}{U(x,s\dot{x}^{\mu})}ds.$
(3.40)
$W_{l}(x,\dot{x})=\frac{U(x,\dot{x})}{4l}\int_{0}^{1}s^{l}\frac{PV_{l-1}(x,s\dot{x}^{\mu})-lPW_{l-1}(x,s\dot{x}^{\mu})}{U(x,s\dot{x}^{\mu})}ds-
lU(x,\dot{x})\int_{0}^{1}s^{l}\frac{V_{l}(x,s\dot{x}^{\mu})}{U(x,s\dot{x}^{\mu})}ds.$
(3.41)

Step 2. Up to now the series defining the function $H$ is a formal series in
$\sigma$, when the coefficients of the differential operator $P$ are analytic,
then the series for $H$ does converge to the exact solution. In our case
however we have smooth coefficients, so the convergence has to be forced by
hand with a suitable cutoff. Let $\rho\in C^{\infty}_{c}(\mathbb{R})$ with
$0\leq\rho\leq 1$, $\rho(t)\equiv 1$ in $|t|\leq 1/2$ and $\rho(t)\equiv 0$ in
$|t|\geq 1$. Suppose that $\\{k_{j}\\}_{j\in\mathbb{N}}$ is a suitable
sequence with $k_{j}\rightarrow\infty$, then claim that

$H(x,y)\doteq\frac{U(x,y)}{\sigma(x,y)}+\sum_{l\geq
0}\rho(\sigma(x,y)k_{l})\left(V_{l}(x,y)\sigma(x,y)^{l}\ln(\sigma(x,y))+W_{l}(x,y)\sigma(x,y)^{l}\right)$
(3.42)

is a parametrix. We are going to show convergence in the Fréchet topology of
$C^{\infty}(\mathcal{O})$ of the series in $\sigma$. First we study the case
where $x\neq y$, when $k_{j}^{-1}\leq\sigma(x,y)$, $\rho=0$, thus, given any
compact subset $K\subset M$, we can estimate

$\sup_{K\times
K}\left|\rho(\sigma(x,y)k_{l})\left(V_{l}(x,y)\sigma(x,y)^{l}+W_{l}(x,y)\sigma(x,y)^{l}\right)\right|\leq\sup_{K\times
K}\left(\frac{1}{k_{l}}\right)^{l}\left(|V_{l}(x,y)||\ln(1/k_{l})|+|W_{l}(x,y)|\right).$

Then to ensure convergence of the series of (3.42) we require that

$\begin{cases}\sum_{l\geq 0}\sup_{K\times
K}\left(\frac{1}{k_{l}}\right)^{l}|V_{l}(x,y)||\ln(1/k_{l})|<\infty,\\\
\sum_{l\geq 0}\sup_{K\times
K}\left(\frac{1}{k_{l}}\right)^{l}|W_{l}(x,y)|<\infty.\end{cases}$

Denote by $\\{k_{j,0}\\}$ the sequence satisfying those estimates. Next, we
study

$\sup_{K\times
K}\left|\nabla_{\mu}\left(\rho(\sigma(x,y)k_{l})\left(V_{l}(x,y)\sigma(x,y)^{l}+W_{l}(x,y)\sigma(x,y)^{l}\right)\right)\right|.$

Then again, we will get some conditions the sequence $\\{k_{j}\\}$ has to
satisfy to ensure that the above quantity is bounded. Denote by
$\\{k_{j,1}\\}$ the sequence obtained. Repeating this at all orders give us
sequences $\\{k_{j,n}\\}$, without loss of generality we can assume that
$\forall j\in\mathbb{N}$, $k_{j,n}\geq k_{j,n-1}$; by a diagonal argument,
define a sequence $k_{j}\doteq k_{j,j}$. With this particular choice of
$\\{k_{j}\\}$, the series defining the parametrix converges uniformly with all
its derivatives to a smooth function in $K\times K\backslash\Delta_{2}(K)$. We
can then take an exhaustion of a compact subset of $M$, for each we determine
the corresponding sequence, then a diagonal argument enables us to conclude
that the series defining $H$ converges with all its derivatives uniformly in
every compact set of $M$.

Step 3. When $y\to x$, or $x,\ y$ are points along a lightlike geodesics,
singularities begin to appear, applying [44, Lemma B.1, Lemma B.2], we obtain
that $H$ is a distribution defined through Hadamard principal value, i.e.

$H(x,y)=\frac{1}{4\pi^{2}}\frac{U(x,y)}{\sigma(x,y)+i\epsilon
t(x,y)+\epsilon^{2}}+\widetilde{V}(x,y)\ln(\sigma(x,y)+i\epsilon
t(x,y)+\epsilon^{2})+\widetilde{W}(x,y);$ (3.43)

where $t$ is the Cauchy temporal function, $t(x,y)\equiv t(x)-t(y)$,
$\epsilon\to 0$ the parameter used to evaluate the principal value of the
integral. ∎

###  3.3 Wick Powers

From the classical results presented in Chapter 2 we would like to enter the
quantum realm. We accomplish this via deformation quantization of the algebra
of microcausal functionals. In this regard, we will assume that the
microcausal algebra of Definition 2.4.6 is maximally defined, that is the
$CO$-open subset $\mathcal{U}$ representing the domain of functionals is the
whole manifold $C^{\infty}(M,\mathbb{R})$. Given a background geometry
$h=(g,m,\kappa)$ we will denote by

$\Big{(}\mathcal{F}_{\mu c}(M,h),\\{\ ,\ \\}_{(M,h)}\Big{)}$

the Poisson algebra of microcausal functional where the background geometry
$h=(g,m,\kappa)$ determines both the wave front set properties (c.f. (2.62))
and the dynamic induced by the Klein-Gordon operator

$\square_{g}+m^{2}+\kappa R(g).$

A deformation quantization of the algebra $\mathcal{F}_{\mu c}(M,h)$ is
therefore an associative algebra with a $\star$ product of the form

$F\star G=\sum_{j}\hbar^{j}\Pi_{j}(F,G)$

for some mappings $\Pi_{j}$ satisfying the consistency conditions

$\displaystyle\Pi_{0}(F,G)$ $\displaystyle=F\cdot G,$
$\displaystyle\Pi_{1}(F,G)-\Pi_{1}(G,F)$ $\displaystyle=i\hbar\langle
dF,\mathcal{G}_{(M,h)}dG\rangle+O(\hbar^{2}),$

where $\cdot$ is the product of the classical algebra defined in (2.19),
$\langle dF,\mathcal{G}_{(M,h)}dG\rangle$ and $\mathcal{G}_{(M,h)}$ are
respectively the Peierls bracket and the causal propagator associated to the
Klein Gordon equations on globally hyperbolic spacetime (c.f. Definition
2.4.4, Theorem 1.2.16). The price we pay to work with such algebras is that
they are infinite series in the deformation parameter $\hbar$.

######  Lemma 3.3.1.

Consider the algebra of regular functionals $\mathcal{F}_{reg}(M,h)$ with the
topology of strong convenient convergence generated by seminorms (2.86). Then