Lennart Ronge

Extracting Hadamard Coefficients from Green’s Operators

Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.)

der

Mathematisch-Naturwissenschaftlichen Fakultät

der

Rheinischen Friedrich-Wilhelms-Universität Bonn vorgelegt von

Lennart Ronge
aus

Flörsheim (Main)

Bonn, 2022

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät

der Rheinischen Friedrich-Wilhelms-Universität Bonn

1\. Gutachter: Prof. Dr. Matthias Lesch
2\. Gutachter: Prof. Alexander Strohmaier

Tag der Promotion: 27.01.2023
Erscheinungsjahr: 2023

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Miscellaneous
      1. 2.1.1 General information
      2. 2.1.2 Motivation: the spectral action
      3. 2.1.3 Some conventions and notation
      4. 2.1.4 Asymptotic expansions
      5. 2.1.5 The Mellin transform
      6. 2.1.6 Binomial coefficients
    2. 2.2 Lorentzian geometry
      1. 2.2.1 Lorentzian vector spaces
      2. 2.2.2 Lorentzian manifolds
    3. 2.3 Function spaces and distributions in vector bundles
    4. 2.4 Green’s operators
    5. 2.5 Riesz distributions and Hadamard coefficients
    6. 2.6 Evolution operators
    7. 2.7 Wavefront calculus
      1. 2.7.1 Wavefront calculus on ${\mathbb{R}}^{n}$
      2. 2.7.2 Wavefront calculus on manifolds
  3. 3 Asymptotic expansions related to Green’s operators
    1. 3.1 Powers of Green’s operators
    2. 3.2 Green’s “resolvent”
  4. 4 Constructing an action-like function
    1. 4.1 Holomorphicity of Riesz distributions for some wavefront
    2. 4.2 Integrating along a timeline
  5. 5 Extracting Hadamard coefficients
    1. 5.1 Hadamard coefficients for $k<\frac{d}{2}$ and $d$ even
    2. 5.2 Arbitrary Hadamard coefficients
  6. 6 Global formulations
    1. 6.1 Global formulation in terms of Green’s operators
    2. 6.2 Spacelike global formulation in terms of evolution operators
  7. Acknowledgements
  8. Appendix: wavefront calculus on manifolds
  9. Notation index
    1. Bibliography

##  Chapter 1 Introduction

Hadamard coefficients are sections in a vector bundle associated to a normally
hyperbolic differential operator in Lorentzian geometry. They are the
Lorentzian equivalent of the heat coefficients of a generalized Laplace
operator (both coefficients are known under various other names, like Seeley-
DeWitt or Schwinger-DeWitt coefficients). While both types of coefficients
arise from the formally same transport equations and are given by the same
formal expressions (up to a factor of $k!$ and in the conventions of [BGV96] a
density factor), they arise in very different contexts: The heat coefficients,
as the name suggests, arise in an asymptotic expansion of the heat semigroup
$e^{-t\Delta}$ for small $t$, while Hadamard coefficients arise in the study
of Green’s operators. Hadamard coefficients occur in quantum field theory on
curved spacetimes (see e.g. [DF08]) and in the Lorentzian version of the APS
index theorem ([BS20]).

Heat coefficients play an important role in various areas of analysis,
geometry and physics. They contain important information about the properties
of the associated Laplace-type operator and the geometry of the underlying
manifold. More information on the heat coefficients and their application to
index theory can be found in [BGV96], for applications in physics see [Vas03].
Their definition via the heat semigroup also makes them available in the
setting of noncommutative geometry, where classical geometric quantities are
no longer available. For example, they can be used to define the scalar
curvature for a noncommutative manifold.

In the Lorentzian case, there is no well-defined heat semigroup and the
Hadamard coefficients need to be defined via recursive differential equations
known as transport equations. These do not make sense in a noncommutative
setting, so the Hadamard coefficients cannot be used there. This thesis is an
attempt to change this by developing a formula for the Hadamard coefficients
(restricted to or integrated over the diagonal) that does not involve local
aspects of the underlying geometry and that might thus be applicable in some
Lorentzian version of noncommutative geometry.

The setting in which Hadamard coefficients originally arise is in the
construction of Green’s operators and in the description of their singularity
structure (see [BGP07, Chapter 2]). More precisely, there is an asymptotic
expansion ([BGP07, Proposition 2.5.1]) in differentiability orders of the form

$\mathcal{K}(G^{\pm})\sim\sum\limits_{k=0}^{\infty}V^{k}R_{\pm}(2k+2),$

where $\mathcal{K}(G^{\pm})$ is the kernel of the advanced/retarded Green’s
operator, the $V^{k}$ are the Hadamard coefficients and the $R_{\pm}(2k+2)$
are the so called Riesz-distributions, which can be thought of roughly as
powers of the distance function. This expansion will serve as the foundation
for this thesis. We want to extract from this a formula for the Hadamard
coefficients on the diagonal in terms of the Green’s operators. As it turns
out, the expansion for a single Green’s operator does not contain enough
information to determine those values, so instead we need to look either at
powers of the Green’s operators or at the analogue of a resolvent.

### Results

Before turning to our formula for Hadamard coefficients, we first obtain some
generalizations of the Hadamard expansion that may be of independent interest.
First, we develop an expansion for powers of the Green’s operator, then an
expansion for the Green’s analogue of a resolvent. Both can be combined in the
following expansion (see 3.2.8)

$\mathcal{K}((G_{P-z}^{\pm})^{m})\sim\sum\limits_{k=0}^{\infty}\binom{m+k-1}{k}V^{k}R_{\pm}(z,2k+2m).$

Here $\mathcal{K}$ refers to taking the Schwartz kernel, $G_{P-z}^{\pm}$ is
the advanced/retarded Green’s operator associated to $P-z$, where
$z\in{\mathbb{C}}$ and $P$ is some normally hyperbolic operator, $V^{k}$ are
the Hadamard coefficients associated to $P$ and $R(z,n)$ are a $z$-dependent
generalization of the Riesz distributions, modeled on fundamental solutions to
$\square-z$ instead of $\square$, which are equal to the ordinary Riesz
distributions in case $z=0$. The asymptotic expansion is up to functions of
arbitrary differentiability.

We then proceed to show our main formula for the Hadamard coefficients. We
obtain the following:

######  Theorem.

(5.2.12) Let $P$ be a normally hyperbolic operator acting on sections of a
vector bundle E over a globally hyperbolic manifold $M$. Let
$o,K\in{\mathbb{N}}$. Let $w$ be a timelike unit speed geodesic in $M$ and
$x=w(0)$. Let $f\in C_{c}^{\infty}({\mathbb{R}})$ be odd and assume that

$\mathcal{M}^{\prime}(f):=\frac{\mathcal{M}(f)}{\Gamma(\frac{\cdot+1}{2})}$

(where $\mathcal{M}$ denotes the Mellin transform) is non-zero on
$\mathbb{Z}$. Let $A:E\otimes E_{x}^{*}\rightarrow{\mathbb{C}}$ be a smooth
fibrewise linear map. Let $V^{K}$ denote the $K$-th Hadamard coefficient of
$P$ and $G_{P-z,x}$ denote the kernel of the causal propagator associated to
$P-z$ for $z\in{\mathbb{C}}$, restricted to the point $x$ in the second
variable. Then we have

$\displaystyle A(V^{K}_{x}(x))=$
$\displaystyle\sum\limits_{m=0}^{K}C(m,K,o,d,f)w^{*}(AG_{P-z,x})[f(\tfrac{\cdot}{s})][[s^{2K+2m+2o-d+3}z^{m+o}]],$

where $[[\cdot]]$ denotes the operation of taking the coefficients in front of
the monomial written in the brackets and

$C(m,K,o,d,f)=\frac{\pi^{\frac{d}{2}-1}4^{K+m+o}(m+o)!K!}{\mathcal{M}^{\prime}(f)({2K+2m+2o-d+3})}\binom{\frac{d}{2}-1-o-K}{m}\binom{2K+o+1-\frac{d}{2}}{K-m}.$

The pull back via $w$ can be thought of as integrating along a timeline. The
map $A$ can be viewed as an arbitrary component function of the Vector bundle
that the Hadamard coefficients are sections of. This is necessary, as vectors
from different fibres cannot be integrated directly.

While this formula is still local, it can be turned into a non-local formula
by multiplying with cut-off functions and integrating. This leads to the
following (using the same setting as in the previous theorem):

######  Theorem.

(6.1.16) Let $\tau$ denote some choice of translation along timelike unit
speed geodesics on $M$. For $\chi_{1},\chi_{2}\in C_{c}^{\infty}(M)$ (with
$\mu_{\chi_{i}}$ denoting the corresponding multiplication operators), there
is $s_{0}>0$, such that for $s<s_{0}$, the operator

$\int\limits_{\mathbb{R}}\mu_{\chi_{1}}f(\tfrac{t}{s})\tau(t)G\mu_{\chi_{2}}dt$

extends continuously to a trace class operator in $L^{2}(E)$ (with respect to
any hermitian structure on $E$). We have for any $K,o\in{\mathbb{N}}$:

$\displaystyle\int\limits_{M}\chi_{1}(x)\chi_{2}(x)\operatorname{tr}(V^{K}_{x}(x))dx$
$\displaystyle=$
$\displaystyle\sum\limits_{m=0}^{K}C(m,K,o,d,f)\operatorname{tr}\left(\int\limits_{\mathbb{R}}\mu_{\chi_{1}}f(\tfrac{t}{s})\tau(t)G\mu_{\chi_{2}}dt\right)[[s^{2K+2m+2o-d+3}z^{m+o}]]$

for $C(m,K,o,d,f)$ as above.

This formula has two main drawbacks.

The first is that the right hand side is fairly complicated. From a
computational point of view, it will generally be more difficult to compute
the kernel of the causal propagator than to just compute the Hadamard
coefficients via the transport equations that define them. For abstract
calculations, the involved combinatorial factors might make it difficult to
equate the right hand side to anything more useful.

The second drawback is the reliance on a notion of time translation along
geodesics, which does not generally exist in a canonical way.

In principle, the formula no longer contains ”local” ingredients. Whether it
is really suitable for a noncommutative setting is hard to answer at the
current time, as the right framework for Lorentzian noncommutative geometry is
still a subject of ongoing research (see [Fra11] for an overview over some
approaches). Such a framework would probably contain some notion of causal
structure, which means it would be likely that the Green’s operators could be
defined as inverses on suitable ”function spaces”. The notion of time
translation is more problematic. Theoretically, something like this could be
defined in the language of noncommutative geometry by choosing a timelike
geodesic unit vector field. Vector fields make sense in noncommutative
geometry as derivations and the condition of being geodesic can be defined in
terms of a covariant derivative, which also exists in the noncommutative
setting. The main problem is that such a vector field may not exist globally
and is not canonical even if it does, even for classical spacetimes.

Such a vector field does exist, however, locally around each point in a
Lorentzian spacetime. Thus one could generally obtain a formula for the
Hadamard coefficients by use of a partition of unity, but this would make the
formula even more complicated and would not really be in the spirit of having
a more global approach.

On the other hand, the fact that the formula works locally makes it likely
that any results obtained by using it that no longer explicitly depend on
$\tau$ would also hold on general globally hyperbolic manifolds and perhaps
also on general Lorentzian noncommutative spaces.

The result is thus to be seen as a tool that might be helpful in finding
further formulas for non-commutative geometry, rather than an alternative way
for actually computing the Hadamard coefficients.

In case the dimension is even, the above difficulties can be avoided for the
first few Hadamard coefficients ($K<\tfrac{d}{2}$). For these, we have a
simpler formula that works for arbitrary timelike curves:

######  Theorem.

(5.1.3) Assume that $d$ is even and $K<\frac{d}{2}$. Assume the above setting,
but now we allow $w$ to be an arbitrary timelike curve with $w(0)=x$ rather
than a unit speed geodesic. Then we have for $k<\tfrac{d}{2}$:

$A_{x}V^{k}_{x}(x)=\frac{(4\pi)^{\frac{d}{2}-1}{(\frac{d}{2}-1-k)}!k!}{\mathcal{M}^{\prime}(f)(1)}w_{x}^{*}(A_{x}G_{P-z,x})[f_{s}][[s^{1}z^{\frac{d}{2}-1-k}]].$

The integrated version of this also holds.

### Structure of proof

In Chapter 2, we will collect some preliminary facts and establish notation.

In Chapter 3, we develop the aforementioned asymptotic expansions for powers
of the Green’s operators and for the Green’s operators associated to the
operators $P-z$ for $z\in{\mathbb{C}}$ (in terms of the Hadamard coefficients
of $P$).

In Chapter 4, we investigate the function given morally by

$L(s):=\int\limits_{\mathbb{R}}f(\tfrac{t}{s})\mathcal{K}(G)(w(t),w(0))dt$

for some timelike curve $w$ and odd $C_{c}^{\infty}$-function $f$. As the
Schwartz kernel of the causal propagator $G=G_{+}-G_{-}$ is a distribution,
the above is not a priori well defined. We thus need to use wavefront calculus
to rigorously define it. We find that $L$ has an asymptotic expansion of the
form (assuming for simplicity that $P$ acts on scalar functions and $w$ is a
unit speed geodesic)

$L(s)\sim\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{\infty}a_{k,n}(V^{k}_{w(0)}\circ
w)^{(2n)}(0)s^{2k+2n+3-d}.$

This is not quite what we want, as the higher time-derivatives of the Hadamard
coefficients introduce error terms (the terms for $n\neq 0$) that prevent us
from extracting the Hadamard coefficients (at $w(0)$) from this asymptotic
expansion.

To remedy this, we need to consider the same expansion for $P-z$ instead of
$P$, which we will do in Chapter 5. This will give us enough additional
information to extract the Hadamard coefficients from the $z$-dependent
version of $L$, which enables us to show the local version of our main
theorem. For the first few Hadamard coefficients for even $d$, this works
fairly easily, while for the others it will be more involved and we will need
$w$ to be a geodesic.

Finally in Chapter 6, we develop the global theorem stated above, as well as
another formulation in terms of evolution operators.

##  Chapter 2 Preliminaries

###  2.1 Miscellaneous

####  2.1.1 General information

We assume that the reader is well acquainted with manifolds and distributions
and has some degree of familiarity with locally convex topological vector
spaces (i.e. working with seminorms). Background information about Lorentzian
geometry and wavefront calculus required for this thesis, as well as a
description of the main objects we will work with (Hadamard coefficients,
Riesz distributions and Green’s operators) and a few other facts will be
provided in this chapter.

An introduction to topological vector spaces and distributions on
${\mathbb{R}}^{d}$ may be found e.g. in [Tre06]. Some information on
Distributions in Vector bundles can be found in the beginning of [BGP07],
which will also be our main reference for the preliminaries on Lorentzian
geometry. Our main source for wavefront calculus will be [BDH16].

In order to help the reader keep track of the various definitions and notation
introduced throughout this thesis (as well as some standard notation that is
being used), a notation index is included at the end. There will also be some
abuse of notation introduced in subsection 2.1.3 that the reader should be
aware of, though most of it is fairly canonical. Much of this is not included
in the notation index, as it consists of omitting notation rather than
introducing it.

The rigorous part of the thesis is occasionally broken by a remark. These are
meant to help the reader understand the motivation and background thought
behind what is happening, without being a part of the logical line of
argumentation, and are thus not as mathematically rigorous as the rest of the
thesis. While any mathematically precise statement in a comment is correct to
the authors best knowledge, it may not be as rigorously checked as the
statements made elsewhere.

####  2.1.2 Motivation: the spectral action

The contents of this subsection are not logically necessary for the subsequent
considerations, but are helpful for understanding the motivation both for the
results and the methods of this thesis.

The (bosonic) spectral action associated to a Dirac Operator $D$ (or an
abstract generalization thereof) is defined as

$\operatorname{tr}(f(sD)),$

where $s>0$ is some cut-off parameter and $f\in C_{c}^{\infty}({\mathbb{R}})$
is some even function that can be chosen freely. This plays an important role,
for example, in the non-commutative geometry based approach to unify the
standard model of particle physics with gravity (see e.g. [CC96]). The full
spectral action also contains another summand that describes fermions, but as
that part is not relevant to the considerations in this thesis, we will use
the term ”spectral action” to refer only to the bosonic part described above.

The spectral action of a Dirac operator $D$ has an asymptotic expansion in
terms of heat coefficients (see [CC96, (2.14)])

$\operatorname{tr}(f(sD))\stackrel{{\scriptstyle s\rightarrow
0}}{{\sim}}\sum\limits_{k=0}^{\infty}c(f,k)a_{k}(D^{2})s^{2k-d},$

where the coefficients $c(f,k)$ are independent of the geometry of the
manifold and $a_{k}(D^{2})$ are the global heat coefficients for the operator
$D^{2}$. The latter are related to the local heat coefficients
$a_{k}(D^{2},x,y)$ (i.e. the Riemannian analogue of the Hadamard coefficients
$V^{k}_{x}(y)$, which are what we are interested in) via

$a_{k}(D^{2})=\int\limits_{M}\operatorname{tr}(a_{k}(D^{2},x,x))dVol(x).$

This asymptotic expansion is not just some curious property of the spectral
action, but actually the reason why it is desirable as an action from a
physical perspective. For example, $a_{1}(D^{2})$ corresponds to the Einstein-
Hilbert action.

As the spacetime we actually live in is Lorentzian rather than Riemannian, it
would be desirable to have this spectral action also for the Lorentzian case.
However, the formula above is not well-defined in this case. A Lorentzian
spectral action for a certain class of spacetimes, where the d’Alembert-
operator is self-adjoint, has been developed in [DW20].

Our aim is to explore the case where the wave operator is no longer self-
adjoint (see [Kam21] for an example of a geodesically complete globally
hyperbolic manifold where this happens) and thus there is no functional
calculus available. The best criterion for an analogue of the spectral action
(which would no longer be ”spectral” in the absence of spectral theory) would
be the existence of an asymptotic expansion similar to the one above, with
Hadamard coefficients instead of local heat coefficients.

As the Lorentzian manifolds one generally considers are not compact, however,
the integral used to define the global coefficients will generally not exist.
Thus the best one can hope for is a corresponding “action density” that could
then be integrated over compact sets. This means, to get an analogue of the
spectral action, one would like a function on the manifold, with extra
parameter $s$, that has an asymptotic expansion

$L(x,s)\stackrel{{\scriptstyle s\rightarrow
0}}{{\sim}}\sum\limits_{k=0}^{\infty}c(f,k)\operatorname{tr}(V_{x}^{k}(x))s^{2k-\delta}$

and is defined purely in terms of functional analytic quantities that could
also be used in noncommutative geometry.

This is both a motivation for and a possible approach toward finding a
functional analytic formula for the (trace of) Hadamard coefficients on the
diagonal (i.e. $V^{k}_{x}(x)$). If one has a formula for Hadamard
coefficients, the postulation of such an expansion might reveal what an
analogue of the spectral action should look like. Conversely, if a function
with the above asymptotic expansion could be defined, one could extract from
this a formula for the Hadamard coefficients by extracting the individual
expansion coefficients. We will not succeed in finding a suitable analogue for
the spectral action in this thesis. However, we will construct a function with
a similar expansion (with additional error terms) in chapter 4. This will
still allow us (though in a more complicated way) to extract a formula for the
Hadamard coefficients.

####  2.1.3 Some conventions and notation

We assume that all manifolds throughout this thesis are imbued with a
canonical volume measure that has a smooth density with respect to the
Lebesgue measure in any coordinate chart (this holds for manifolds with smooth
metrics, which really are the only cases we care about). For these we will
write $dx$ instead of $d\operatorname{Vol}(x)$. As usual, we will identify a
locally integrable function $f$ on a space $X$ with a canonical volume measure
with the distribution $T_{f}$ given by

$T_{f}(\phi)=\int\limits_{X}\phi fd\operatorname{Vol}.$

In order to avoid confusion from this abuse of notation, we will write
evaluation of distributions with square brackets:

$\eta[\phi]:=\eta(\phi)$

for some distribution $\eta$. By our identification, we thus denote

$f[\phi]:=T_{f}(\phi)$

for any locally integrable function $f$.

Complex numbers will sometimes be identified with the corresponding multiple
of the identity or the constant function with that value.

Occasionally, we will use $C$ or $C^{\prime}$ without further specification.
These will denote arbitrary constants, that may take new values for each new
equation.

We will often encounter definitions and statements that work in both
directions of time symetrically, usually distinguished by an index $+$ or $-$.
In order to avoid writing everything twice, we use the symbol $\pm$ to denote
both cases at once. The use of $\pm$ or $\mp$ in a definition or theorem
indicates that this should hold both with the upper sign chosen everywhere or
with the lower sign chosen everywhere in that definition or theorem. In some
cases we also need to change the wording depending on the choice of time
alignment. In that case we will use a ”/” to indicate that the first word is
to be used with the upper sign and the second word is to be used with the
lower sign.

For example, the statement ”If $A$ is past/future compact, $J_{\pm}(A)\cap
J_{\mp}(x)$ is compact.” means ”If $A$ is past compact, $J_{+}(A)\cap
J_{-}(x)$ is compact and if $A$ is future compact, $J_{-}(A)\cap J_{+}(x)$ is
compact.”.

We introduce some identifications and abuse of notations for vector bundles:

######  Fixed Notation 2.1.1.

Throughout this thesis, for any vector spaces $V$, $W$, $U$ (mostly, these
will be fibres of vector bundles), we will canonically identify:

  * •

${\mathbb{C}}^{*}$ with ${\mathbb{C}}$ (identifying $\lambda Id$ with
$\lambda$)

  * •

$V^{**}$ with $V$

  * •

$V\otimes{\mathbb{C}}$ with $V$

  * •

$(V\otimes W)^{*}$ with $V^{*}\otimes W^{*}$

  * •

$W\otimes V^{*}$ with $Hom(V,W)$

  * •

$(V\otimes W)\otimes U$ with $V\otimes(W\otimes U)$

  * •

$V\otimes W$ with $W\otimes V$ if $V$ and $W$ are distinct and this
identification is required for definitions to make sense.

Moreover, we omit composition and evaluation of dual vectors and fibrewise
linear maps from the notation, so that e.g. for $v\in V$, $w\in W^{*}$ and
$A\in Hom(V,W)\cong W\otimes V^{*}\cong((W^{*})\otimes V)^{*}$, we write:

$wAv=w(A(v))=A(w\otimes v).$

We identify scalar functions on some manifold $X$ with sections of the vector
bundle $X\times{\mathbb{C}}$.

The main purpose of most of this abuse of notation is to be able to ”multiply”
sections in suitable vector bundles without having to define a zoo of
canonical bilinear maps and having to do case distinctions for all of them.

We introduce notation for Schwartz kernels (or integral kernels) of operators:

######  Definition 2.1.2.

Let $E$ anf $F$ be vector bundles. For a continuous linear operator
$T\colon\Gamma_{c}(E)\rightarrow\mathcal{D}^{\prime}(F)$, we denote by
$\mathcal{K}(T)$ its Schwartz kernel, i.e. the unique distribution in
$F\boxtimes E^{*}$ such that for any test functions $\psi\in\Gamma_{c}(F^{*})$
and $\phi\in\Gamma_{c}(E)$, we have

$\mathcal{K}(T)[\psi\otimes\phi]=T(\phi)[\psi].$

The Schwartz kernel theorem asserts that this is well-defined.

######  Fixed Notation 2.1.3.

For some space of sections or distributions $D(E)$ in some vector bundle $E$
and a finite dimensional vector space $W$, we identify the corresponding space
of sections/distributions $D(E\otimes W)$ with $D(E)\otimes W$. For a linear
operator $T$ acting on $D(E)$, $\eta\in D(E)$ and $w\in W$, we define

$T(\eta\otimes w):=T(\eta)\otimes w,$

i.e. we identify $T$ with $T\otimes id$. This corresponds to applying $T$ to
every ”component” with respect to $W$.

This will be needed because we will often have to deal with distributions in
$E\otimes E^{*}_{x}$ for some vector bundle $E$ and $x\in M$, which correspond
to Schwartz kernels of operators in $E$, restricted to $x$ in the second
component. Application in the first component as defined here then corresponds
to composition of operators.

####  2.1.4 Asymptotic expansions

An important notion in the following will be that of an asymptotic expansion.
It is customarily written with an infinite summation sign, even though the sum
as written may not converge (at least not in a usual topology - one could
construct non-Hausdorff topologies, in which these asymptotic expansions
actually correspond to limits). There will be two types of asymptotic
expansions in this thesis: Asymptotic expansions in a parameter, where the sum
gives an approximation up to arbitrary powers of some expansion parameter $s$
and asymptotic expansions in differentiability orders, where the approximation
is up to arbitrarily differentiable functions. In the latter case, we will
also need extra notation for the case where one argument is fixed. The basic
idea in all cases is that finite partial sums of the right hand side
approximate the left hand side up to arbitrary ”order”, which is given either
by decay in a parameter or by differentiability.

######  Definition 2.1.4.

  * •

Assume that $F$ and $f_{n}$ for $n\in{\mathbb{N}}$ are functions from an open
interval containing $0$ to ${\mathbb{C}}$. We write

$F(s)\stackrel{{\scriptstyle s\rightarrow
0}}{{\sim}}\sum\limits_{n=0}^{\infty}f_{n}(s)$

if and only if for any $m\in{\mathbb{N}}$, there is $N_{0}\in{\mathbb{N}}$,
$C\in{\mathbb{R}}$ and $\epsilon>0$ such that for $|s|<\epsilon$ and $N\geq
N_{0}$, we have

$\Big{|}F(s)-\sum\limits_{n=0}^{N}f_{n}(s)\Big{|}\leq Cs^{m}.$

  * •

Assume that $F$ and $f_{n}$ for $n\in{\mathbb{N}}$ are distributions on a
manifold $X$. We write

$F\sim\sum\limits_{n=0}^{\infty}f_{n}$

if and only if for any $m\in{\mathbb{N}}$, there is $N_{0}\in{\mathbb{N}}$,
such that we have for $N\geq N_{0}$

$F-\sum\limits_{n=0}^{N}f_{n}\in C^{m}(X).$

  * •

Let $X$ and $Y$ be manifolds and assume that $F(x)$ and $f_{n}(x)$ for
$n\in{\mathbb{N}}$ are distributions on $Y$ for any $x\in X$. We write

$F(x)\sim_{x}\sum\limits_{n=0}^{\infty}f_{n}(x)$

if and only if for any $m\in{\mathbb{N}}$, there is $N_{0}\in{\mathbb{N}}$ and
$R\in C^{k}(Y\times X)$, such that we have for all $x\in X$ and $N\geq N_{0}$

$F(x)-\sum\limits_{n=0}^{N}f_{n}(x)=R(\cdot,x).$

In each case, iterated sums are defined accordingly: For every $m$,
sufficiently large partial sums of the right hand side should approximate the
left hand side up to any differentiability order or decay order in $s$.

Asymptotic expansions are robust under reordering or repackaging the summands:

######  Proposition 2.1.5.

Let $\sim_{*}$ be any of the three asymptotic expansion types defined above.

  1. 1.

Whether an asymptotic expansion holds only depends on the set of summands on
the right hand side, not on the order of their summation.

  2. 2.

Assume that $A$, $A_{k}$ and $a_{kl}$ are elements in some space where
$\sim_{*}$ makes sense. Assume furthermore that

$A_{k}\sim_{*}\sum\limits_{l=0}^{\infty}a_{kl}.$

Then we have

$A\sim_{*}\sum\limits_{k,l=0}^{\infty}a_{kl}$

if and only if

$A\sim_{*}\sum\limits_{k=0}^{\infty}A_{k}.$

###### Proof sketch.

Let $m\in{\mathbb{N}}$ be arbitrary. If any asymptotic expansion holds in one
of the above cases, all but finally many summands involved must be $C^{m}$ or
$O(s^{m})$ (depending on the type of asymptotic expansion), since otherwise we
could not have equality up to $C^{m}$ resp. $O(s^{m})$ for arbitrarily large
partial sums. Thus, to check that the expansion holds up to degree $m$, it
suffices to consider only finitely many summands. As the statements above hold
for finite sums, they also hold for asymptotic expansions. ∎

####  2.1.5 The Mellin transform

######  Definition 2.1.6.

For a function $f\colon(0,\infty)\rightarrow{\mathbb{C}}$, define its Mellin
transform via

$\mathcal{M}(f)(\alpha):=\int\limits_{0}^{\infty}f(x)x^{\alpha-1}dx$

for any $\alpha$ such that the integral exists. If $f$ is defined on a larger
domain, its Mellin transform is defined as that of its restriction to
$(0,\infty)$.

This is the analogue of the Fourier transform when identifying ${\mathbb{R}}$
with $(0,\infty)$ via the exponential map. For $f$ compactly supported in
$(0,\infty)$, the integral always exists for any $\alpha$ and is holomorphic
as a function of $\alpha$. For $f\in C_{c}^{\infty}({\mathbb{R}})$, the
integral is still defined and holomorphic if $\Re(\alpha)>0$. We define it for
$\Re(\alpha)<0$ by meromorphic continuation:

######  Definition/Proposition 2.1.7.

For $f\in C_{c}^{\infty}(M)$, the Mellin transform of $f$ admits a meromorphic
extension to ${\mathbb{C}}$ with simple poles at most at negative integers
(including zero), whose residues are given by

$\operatorname{Res}\limits_{\alpha=-k}\mathcal{M}(f)(\alpha)=\frac{(-1)^{k}}{k!}f^{(k)}(0)$

and no other poles. We also denote this extension by $\mathcal{M}(f)$.

######  Remark 2.1.8.

There is a more general connection between asymptotic expansions of a function
at zero or infinity and the pole structure of its Mellin transform (see e.g.
Proposition 2.1.2 and the preceding paragraph in [Les96] or [FGD95, Theorem
3]).

###### Proof.

For $\Re(\alpha)>0$, partial integration yields

$\mathcal{M}(f^{\prime})(\alpha+1)=-\alpha\mathcal{M}(f)(\alpha).$

Using the left hand side of this equality as a definition for the right hand
side, one can successively extend $\mathcal{M}(f)$ to the left
meromorphically, with poles only at negative integers arising from the case
$\alpha=0$. Taking the limit as $\alpha$ aproaches zero of the above equality
and using the fundamental theorem of calculus, we obtain

$\operatorname{Res}_{\alpha=0}\mathcal{M}(f)(\alpha)=-\mathcal{M}(f^{\prime})(1)=-\int\limits_{0}^{\infty}f^{\prime}(x)dx=f(0).$

Applying the previous equality iteratively (by holomorphic continuation, it
holds for arbitrary values of $\alpha$), we obtain

$\operatorname{Res}_{\alpha=-k}\mathcal{M}(f)(\alpha)=\frac{(-1)^{k}}{k!}\operatorname{Res}_{\alpha=0}\mathcal{M}(f^{(k)})(\alpha)=\frac{(-1)^{k}}{k!}f^{(k)}(0).\qed$

####  2.1.6 Binomial coefficients

We will need to use binomial coefficients for arbitrary complex arguments.
Those are defined in terms of the Gamma function:

$\binom{\alpha}{\beta}:=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}$

for any $\beta\in{\mathbb{C}}$ and
$\alpha\in{\mathbb{C}}\backslash-{\mathbb{N}}$. In case both $\alpha$ is a
negative integer and $\beta$ is also an integer, we define the binomial
coefficient via continuous extension in $\alpha$, leaving $\beta$ fixed:

$\binom{\alpha}{\beta}:=\lim\limits_{\alpha^{\prime}\rightarrow\alpha}\binom{\alpha^{\prime}}{\beta}.$

This is holomorphic in $\alpha$ for fixed $\beta$. A consequence of this is
that $\binom{\alpha}{\beta}$ vanishes whenever $\beta$ is a negative integer.

###  2.2 Lorentzian geometry

In this section we collect the facts about Lorentzian geometry that will be
required for this thesis. More information on Lorentzian geometry can be found
e.g. in [O’N83].

####  2.2.1 Lorentzian vector spaces

We will first consider basic notions in Lorentzian vector spaces, before
extending them to general Lorentzian manifolds.

######  Definition 2.2.1.

A Lorentzian bilinear form on a d-dimensional vector space is a symmetric
bilinear form $\eta$ with signature $(d-1,1)$. A Lorentzian vector space is a
vector space with a Lorentzian bilinear form.

The basic example of a Lorentzian vector space is Minkowski space:
${\mathbb{R}}^{d}$ with the Lorentzian bilinear form

$\eta(x,y)=-x_{0}y_{0}+\sum\limits_{i=1}^{d-1}x_{i}y_{i}$

(the indexing convention is from relativistic physics, where time gets the
index $0$). Every Lorentzian vector space is isometric to Minkowski space
(more or less directly by the definition of the signature), so most theorems
only need to be shown for Minkowski space. Eventually, they will usually be
applied to tangent spaces of a Lorentzian manifold.

In the following, let $V$ be a Lorentzian vector space with bilinear form
$\eta$. The bilinear form singles out a “general direction” that has a
different sign than the others (a negative sign with the above definition, but
there are different sign conventions). This direction is regarded as “time”,
the others as “space”, as made precise in the following definition:

######  Definition 2.2.2.

A vector $v\in V\backslash\\{0\\}$ is called

  * •

timelike, if $\eta(v,v)<0$

  * •

spacelike, if $\eta(v,v)>0$

  * •

lightlike, if $\eta(v,v)=0$

  * •

causal, if $\eta(v,v)\leq 0.$

$0$ is considered as spacelike.

The Minkowski-orthogonal complement of a timelike vector is spacelike:

######  Lemma 2.2.3.

If $x,y\in V$ satisfy $\eta(x,y)=0$ and $x$ is timelike, then $y$ is
spacelike. If instead $y\in V^{*}$ and $y(x)=0$, then $y$ is spacelike with
respect to the induced metric on $V^{*}$.

###### Proof.

Without loss of generality, assume that $V$ is Minkowski space. Let
$x=(x_{0},x_{r})$ and $y=(y_{0},y_{r})$. $x$ being timelike is equivalent to
$|x_{0}|>\|x_{r}\|$. We have

$|\eta(x,y)|=|-x_{0}y_{0}+\left<x_{r},y_{r}\right>|\geq|x_{0}||y_{0}|-\|x_{r}\|\|y_{r}\|\geq|x_{0}|(|y_{0}|-\|y_{r}\|),$

with equality only if $y_{r}=0$. Thus if the left hand side is zero, we either
have $y=0$ or $|y_{0}|-\|y_{r}\|<0$, which means that $y$ is spacelike. This
concludes the proof of the first statement.

If $y$ is an element of the dual space, it is induced by some $y^{\sharp}\in
V$ via

$y(w)=\eta(y^{\sharp},w).$

The induced form on the dual is defined so that

$\eta(y,y):=\eta(y^{\sharp},y^{\sharp}),$

so $y$ is spacelike if and only if $y^{\sharp}$ is. If

$\eta(y^{\sharp},x)=y(x)=0,$

this is the case by the previous part. ∎

The causal vectors form two cones, known as the (solid) light cones.

######  Proposition 2.2.4.

The set of causal vectors in $V$ has two connected components, each of which
is convex. If $V$ is Minkowski space, these consist of the vectors with
positive/negative $0$-component. A timelike vector $x$ and a causal vector $y$
are in the same connected component if and only if $\eta(x,y)<0$.

###### Proof.

Without loss of generality, assume that $V$ is Minkowski space. Let
$x=(x_{0},x_{r})$ and $y=(y_{0},y_{r})$ be causal vectors with
$x_{0},y_{0}>0$. We have

$\eta(x,y)=-x_{0}y_{0}+\left<x_{r},y_{r}\right>\leq-
x_{0}y_{0}+\|x_{r}\|\|y_{r}\|\leq-x_{0}y_{0}+x_{0}y_{0}=0$

and thus for $a,b\in(0,1)$

$\eta(ax+by,ax+by)=a^{2}\eta(x,x)+b^{2}\eta(y,y)+2ab\eta(x,y)<0.$

Thus the set of causal vectors with positive $0$-component is convex and hence
connected. The same holds for those with negative $0$ component, because $-x$
is causal if and only if $x$ is. As there are no causal vectors with
$x_{0}=0$, those two sets are disconnected from each other. This proves the
first two statements.

If $x$ is timelike, Lemma 2.2.3 implies the function $\eta(x,\cdot)$ does not
take the value 0 on the set of causal vectors. As it is also continuous, the
preimage of $(0,\infty)$ and $(-\infty,0)$ must be both open and closed. As
neither is empty, since they contain $-x$ resp. $x$, they must be the two
connected components decribed above. Thus the connected component of $x$
consists of all those $y$ with $\eta(x,y)<0$. ∎

In many cases, it is desirable to be able to have a time direction, i.e. to be
able to talk about future and past. This is called a time-orientation. Rather
than choosing a specific vector as the future direction, one only specifies
which light cone points into the future and which one points into the past.

######  Definition 2.2.5.

A time-orientation on a Lorentzian vector space $V$ is a choice of one
connected component of the set of causal vectors. Vectors in this component
will be called future oriented, vectors in the other component will be called
past oriented. On Minkowski space, one canonically takes those vectors with
positive $0$-component to be future-directed. In the following, we will assume
that all Lorentzian vector spaces are time-oriented.

####  2.2.2 Lorentzian manifolds

We now get to Lorentzian manifolds. These are manifolds where each tangent
space is a Lorentzian vector space.

######  Definition 2.2.6.

A Lorentzian manifold is a smooth manifold $M$ (of dimension $d$) with a
smooth $(0,2)$-tensor field $g$ that is a Lorentzian bilinear form on each
fibre $T_{x}M$. g is called a Lorentzian metric.

In order to talk about future and past, one has to choose a time-orientation
at each point. A timelike vector field naturally defines such a time-
orientation in a continuous way. This does not exist on arbitrary Lorentzian
maniflolds, so we have to assume its existence. Note that this vector field is
far from unique or canonical, but only used to ensure the choice of
orientations is continuous.

######  Definition 2.2.7.

A continuous timelike vector field $\tau$ on $M$ induces a time-orientation on
each tangent space: A causal vector $v\in T_{x}M$ is called future oriented
(with respect to $\tau$), if $g(v,\tau(x))<0$ and past oriented otherwise. A
time-orientation on $M$ is a collection of time-orientations on each $T_{x}M$
that arises in this way. $M$ is called time-orientable, if a time-orientation
exists, and time-oriented, if one has been chosen.

######  Fixed Notation 2.2.8.

Throughout this paper, $M$ will be a time-oriented Lorentzian manifold of
dimension $d\geq 2$ with metric $g$.

We will later additionally assume that $M$ is globally hyperbolic (once we
have defined what that means).

The definitions made above for vectors can be extended to curves by looking at
their tangent vectors:

######  Definition 2.2.9.

A piecewise smooth curve $\gamma$ is called future/past oriented, if the same
is true for every tangent vector $\gamma^{\prime}(t)$. A future or past
oriented curve is called timelike/lightlike/causal if the same is true for
every tangent vector.

Using this, one can also talk about the future and past of a point $p\in M$,
i.e. all points that can be reached from $p$ via a future/past oriented curve:

######  Definition 2.2.10.

The causal future/past of $A\subset M$, denoted by $J_{\pm}(A)$, is the set of
all points $y\in M$. such that there is a future/past oriented causal curve
$\gamma\colon[s,t]\rightarrow M$ (for $s,t\in{\mathbb{R}}$) with $\gamma(s)\in
A$ and $\gamma(t)=y$. The causal future/past of a point $x\in M$ is that of
the corresponding one-point set

$J_{\pm}(x):=J_{\pm}(\\{x\\}).$

Set $J(A):=J_{+}(A)\cup J_{-}(A)$. Here we allow trivial curves, i.e. $x\in
J_{\pm}(x)$.

This causal structure is central to Lorentzian geometry and will be used
frequently throughout this thesis. Whenever it is unclear in which manifolds
this is done, the manifold will be indicated as a superscript. We observe that
$J_{\pm}(J_{\pm}(A))=J_{\pm}(A)$, as we can concatenate future/past directed
curves.

An important condition we have to impose on our manifold is that of global
hyperbolicity. This means that there is a surface, called a Cauchy
hypersurface, that can be thought of as all of space at one instant of time.

######  Definition 2.2.11.

A Cauchy hypersurface in $M$ is a hypersurface that is intersected exactly
once by any inextensible timelike curve.

A connected timeoriented Lorentzian manifold is called globally hyperbolic, if
it contains a Cauchy hypersurface.

One example of a globally hyperbolic spacetime is Minkowski space. There are
several equivalent ways of characterizing global hyperbolicity (see [BGP07,
Theorem 1.3.10])

######  Theorem 2.2.12.

The following are equivalent:

  1. 1.

For any $x,y\in M$, $J_{+}(x)\cap J_{-}(y)$ is compact and every point in $M$
has a neighborhood basis of open sets $V$ such that every causal curve with
endpoints in $V$ lies entirely in $V$ (the second part is known as strong
causality).

  2. 2.

$M$ contains a Cauchy hypersurface.

  3. 3.

There is a $d-1$ dimensional manifold $\Sigma$, a family of Riemannian metrics
$(g_{t})_{t\in{\mathbb{R}}}$ on $\Sigma$ and a smooth positive function
$\beta$ on $\Sigma\times{\mathbb{R}}$ such that $M$ is isometric to
$\Sigma\times{\mathbb{R}}$ with metric $g_{t}\oplus-\beta dt^{2}$. This can be
chosen so that each $\Sigma\times\\{t\\}$ is a smooth spacelike Cauchy
hypersurface.

######  Fixed Notation 2.2.13.

From now on, we will always assume that $M$ is globally hyperbolic.

One consequence is that no point is in the interior of its own future or past:

######  Proposition 2.2.14.

Each $x\in M$ is in the topological boundary of $J(x)$.

###### Proof.

$x$ is always contained in $J(x)$. By Theorem 2.2.12, $x$ is contained in some
Cauchy hypersurface $\Sigma$. As this has dimension $d-1\geq 1$, there is a
sequence in $\Sigma\backslash\\{x\\}$ that converges to $x$. As every causal
curve through $x$ cannot intersect $\Sigma$ in any other point (by definition
of a Cauchy hypersurface), that sequence is in the complement of $J(x)$, so we
can conclude that $x$ is in its boundary. ∎

We will often need to talk about sets that are compact in the past/future:

######  Definition 2.2.15.

A set $A\subseteq M$ is called past/future compact, if for any $x\in M$, the
intersection $A\cap J_{\mp}(x)$ is compact. It is called strictly past/future
compact, if it is closed and contained in the future/past of a compact set.

Strictly past/future compact sets are indeed, as the name suggests,
past/future compact. Moreover, one may replace the point in the definition
with an arbitrary compact set.

######  Proposition 2.2.16.

If $K$ and $K^{\prime}$ are compact, then $J_{+}(K)\cap J_{-}(K^{\prime})$ is
compact. In particular, every strictly past/future compact set is past/future
compact.

(see Lemma A.5.7 in [BGP07], with the second statement following by inserting
a point for $K^{\prime}$ resp. $K$)

######  Proposition 2.2.17.

If $A$ is past/future compact and $K$ is compact, then

$A\cap J_{\mp}(K)$

is compact.

(see Theorem 3.1 in [San13])

One further definition that we will need is that of causal compatibility,
which essentially states that the causal structure of a subset is compatible
with that of the manifold:

######  Definition 2.2.18.

An open subset $U$ of $M$ is called causally compatible, if for any
$A\subseteq U$

$J_{\pm}^{U}(A)=J_{\pm}^{M}(A)\cap U,$

where superscripts on $J_{\pm}$ indicate in which manifold the future/past is
taken.

In other words, any two points in $U$ that can be connected by a causal curve
in $M$ can also be connected by a causal curve in $U$.

As in the Riemannian case, one can define a Levi-Civita connection in the
Lorentzian case and use this to define geodesics and exponential maps. The
exponential map is particularly useful on subsets where they are
diffeomorphisms

######  Definition 2.2.19.

An open subset $U\subseteq M$ is called geodesically convex (or convex, for
short), if the exponential map defines a diffeomorphism from an open
neighborhood of the zero section of $TU$ to $U\times U$.

Every point in $M$ has a neighborhood basis of convex subsets. For such $U$,
the exponential map defines a diffeomorphism from an open subset of $TU$ to
$U\times U$. The exponential map also preserves causality in the following
sense (see [HE73, Proposition 4.5.1]111The book works with $d=4$, as it is
motivated by physical applications, but this is not used anywhere in the
proof.):

######  Proposition 2.2.20.

Let $U\subseteq M$ be convex. Any $x,y\in U$ that can be joined by a
future/past oriented causal (resp. timelike) curve can also be joined by a
future/past oriented causal (resp. timelike) geodesic in $U$. This means that

$J_{\pm}(x)=exp_{x}(J_{\pm}(0)).$

###  2.3 Function spaces and distributions in vector bundles

We now define spaces of functions and distributions that will be used
throughout the paper. Readers will probably be familiar with at least some of
them, but we also want to fix notation. Let $E$ be a vector bundle over a
manifold $X$. Let $\kappa$ be a locally finite collection of charts covering
$X$ and $\tau$ a locally finite collection of local trivializations.

######  Definition 2.3.1.

For a compact set $K\subseteq X$, and $n\in{\mathbb{N}}$ , we define
$C^{n}$-seminorms on $n$ times continuously differentiable sections by

$\|f\|_{C^{n}(K)}:=\sup\\{\|\Psi\circ
f|_{K}\circ\phi^{-1}\|_{C^{n}}\mid\phi\in\kappa,\Psi\in\tau\\}.$

Here we use the convention that all functions in a composition automatically
have their domain restricted to those elements on which the composition is
defined.

In general, this depends on the choice of $\tau$ and $\kappa$, but the
seminorms for different choices are equivalent. As we do not care about
numerical values in this thesis, we can thus assume that we have chosen a
locally finite collection of charts and local trivializations for each vector
bundle that will be considered in this thesis and interpret all $C^{k}$-norms
as being taken with respect to those.

The following spaces are fairly standard:

######  Definition 2.3.2.

Let $E$ be a vector bundle over a manifold $X$. For each
$k\in{\mathbb{N}}\cup\\{\infty\\}$, let $\Gamma^{k}(E)$ be the space of all
$k$ times continuously differentiable sections with topology induced by the
seminorms $\|\cdot\|_{C^{n}(K)}$ for all compact subsets $K\subseteq M$ and
$n=k$ in case $k\in{\mathbb{N}}$ or all $n\in{\mathbb{N}}$ in case $k=\infty$.

For $A\subseteq X$, let $\Gamma^{k}_{A}(E)$ be those sections whose support is
contained in $A$, with the subspace topology.

Let $\Gamma^{k}_{c}(E)$ be the space of all compactly supported
$C^{k}$-sections, with the limit topology obtained from the spaces
$\Gamma^{k}_{K}(E)$ for $K\subseteq X$ compact.

In case $k=\infty$, we may omit the superscript and just write $\Gamma$
instead of $\Gamma^{\infty}$.

Somewhat less well-known variants in Lorentzian geometry are the following
spaces:

######  Definition 2.3.3.

Let $E$ be a vector bundle over $M$. For $k\in{\mathbb{N}}\cup\\{\infty\\}$,
let $\Gamma^{k}_{\pm}(E)$ be the space of past/future compactly supported
$C^{k}$-sections, with the limit topology obtained from all spaces
$\Gamma^{k}_{A}(E)$ for $A\subseteq M$ past/future compact.

Let $\Gamma^{k}_{s\pm}(E)$ be the space of strictly past/future compactly
supported $C^{k}$-sections, with the limit topology obtained from all spaces
$\Gamma^{k}_{A}(E)$ for $A\subseteq M$ strictly past/future compact.

Again, we may omit $k$ if it is $\infty$.

As a special case of the above definitions, when $E=M\times{\mathbb{C}}$, we
obtain spaces of complex valued functions, in which case we will write
$C^{k}_{\cdot}(M)$ instead of $\Gamma^{k}_{\cdot}(E)$. Taking duals, we obtain
spaces of distributions:

######  Definition 2.3.4.

Define the following spaces of distributions in a vector bundle $E$ over $M$:

  * •

The space of distributions $\mathcal{D}^{\prime}(E)$ is the space of
continuous linear maps from $\Gamma_{c}(E^{*})$ to ${\mathbb{C}}$.

  * •

The space of past/future compactly supported distributions
$\mathcal{D}^{\prime}_{\pm}(E)$ is the space of continuous linear maps from
$\Gamma_{s\mp}(E^{*})$ to ${\mathbb{C}}$.

  * •

The space of strictly past/future compactly supported distributions
$\mathcal{D}^{\prime}_{s\pm}(E)$ is the space of continuous linear maps from
$\Gamma_{\mp}(E^{*})$ to ${\mathbb{C}}$.

On each of these spaces, we use the topologies induced by the seminorms

$\|\eta\|_{B}:=\sup\limits_{\phi\in B}\|\eta(\phi)\|$

for all bounded subsets of the corresponding test space.

######  Remark 2.3.5.

Instead of the (strong) topology used here, the weak*-topology is often used
on spaces of distributions. As both topologies induce the same convergence of
sequences (and thus also the same notion of holomorphicity), they are
essentially equivalent for the deliberations in this thesis. The reason we
will use the strong topology is that when we get to wavefront calculus, this
will make certain operations continuous rather than just sequentially
continuous, which makes this choice of topology seem more natural.

As the names would suggest, the space of (strictly) past/future compactly
supported distributions is in canonical bijection with distributions whose
support is (strictly) past/future compact, via restricting to $\Gamma_{c}(E)$
(see [Bär14, Lemma 2.13]).

Moreover, we have (see [Bär14, Lemma 2.15]):

######  Proposition 2.3.6.

Smooth compactly supported sections are (sequentially) dense in any of the
above spaces of distributions.

There are two ways of transporting distributions from one space to another via
some function $f$: pullback and pushforward

######  Definition 2.3.7.

Let $f\colon X\rightarrow Y$ be a map between manifolds.

For a function $g$ on $Y$, define the pullback by

$f^{*}g:=g\circ f.$

For a compactly supported distribution $\eta$ on $X$, define the pushforward
by

$f_{*}\eta[\phi]:=\eta[\phi\circ f]$

for every testfunction $\phi$. If $E$ is a vector bundle on $Y$, these also
define maps

$f^{*}\colon\Gamma(E)\rightarrow\Gamma(f^{*}(E))$

and

$f_{*}\colon\Gamma(f^{*}(E))\rightarrow\Gamma(E).$

Neither of the two operations is defined on arbitrary distributions for
general $f$: The pullback requires the existence of a continuous extension,
while the pushforward requires $f$ to be proper on the support of the
argument. Both will be well defined for all distributions in case $f$ is a
diffeomorphism. Note that, even then, $f^{*}$ and $f^{-1}_{*}$ will not
generally coincide, since the pullback of a distribution depends on the choice
of canonical volume, which may not be preserved by $f$.

For sufficiently regular kernels, the trace of an operator is given by
integrating its kernel over the diagonal:

######  Theorem 2.3.8.

If $T$ is an operator acting on sections of a vector bundle $E$ over a
manifold $X$ whose Schwartz kernel is smooth and compactly supported, then $T$
is trace class in $L^{2}(E)$ (with respect to any hermitian metric on $E$) and

$\operatorname{tr}(T)=\int\limits_{X}\operatorname{tr}(\mathcal{K}(T)(x,x))dx.$

###### Proof.

[DR14, Theorem 1.1] asserts a stronger statement in the case of operators
acting on scalar functions in a closed manifold $M$. This can be extended to
the case at hand. First, choose a precompact open set with smooth boundary
such that $\mathcal{K}(T)$ is supported in $U$. Consider the double of
$\bar{U}$, i.e. two copies of $\bar{U}$ glued along $\partial U$ (see, for
example [Kos92, Section VI.5]). This gives us a closed manifold containing $U$
as a subset. As everything outside of $U$ is irrelevant for the mapping
propeties of $T$, this reduces the theorem to the case where $X$ is compact.

By Swan’s Lemma, every vector bundle over a compact manifold is a direct
summand of a trivial bundle (see [Swa62, Corollary 5]). Choosing a Hermition
structure on the other summand and using the direct sum of the Hermitian
structure on the trivial bundle (which is then isometric to $X\times R^{N}$),
we obtain that $L^{2}$-sections in $E$ are isometrically embedded into those
in $X\times{\mathbb{R}}^{N}$. All other function spaces are also embedded in a
compatible way. As traces are preserved by isometric embeddings, it suffices
to consider the case $E=X\times{\mathbb{R}}^{N}$.

Let $p_{i}$ and $\iota_{i}$ denote composition with the projection on and the
inclusion in the $i$-th component of $X\times{\mathbb{R}}^{N}$ (on
distributions, $p_{i}$ can be defined by applying $\iota_{i}$ to the test
function). Consider the components

$T_{ij}:=p_{i}\circ T\circ\iota_{j}.$

This has Schwartz kernel

$\mathcal{K}(T_{ij})=\mathcal{K}(T)_{ij},$

as

$\mathcal{K}(T)_{ij}[\phi\otimes\psi]=\mathcal{K}(T)[\iota_{i}\phi\otimes\iota_{j}\psi]=p_{i}T\iota_{j}(\psi)[\phi]=\mathcal{K}(T_{ij})[\phi\otimes\psi].$

As the claim holds for scalar functions, we can conclude that $T_{ij}$ defines
a trace class operator with trace

$\displaystyle\operatorname{tr}(T_{ij})$
$\displaystyle=\int\limits_{X}\mathcal{K}(T_{ij})(x,x)dx$
$\displaystyle=\int\limits_{X}\mathcal{K}(T)_{ij}(x,x)dx.$

As the $\iota_{i}$ are isometric embeddings into mutually orthogonal subspaces
(and $p_{i}$ are the corresponding projections), we have

$\operatorname{tr}(\iota_{i}T_{ij}p_{j})=\delta_{ij}\operatorname{tr}(T_{ii}).$

We can thus conclude that

$T=\sum_{i,j=1}^{N}\iota_{i}T_{ij}p_{j}$

is trace class with

$\displaystyle\operatorname{tr}(T)$
$\displaystyle=\sum\limits_{i,j=1}^{N}\delta_{ij}\operatorname{tr}(T_{ii})$
$\displaystyle=\sum\limits_{i=1}^{N}\operatorname{tr}(T_{ii})$
$\displaystyle=\sum\limits_{i=1}^{N}\int\limits_{X}\mathcal{K}(T)_{ii}(x,x)dx$
$\displaystyle=\int\limits_{X}\operatorname{tr}(\mathcal{K}(T)(x,x))dx.\qed$

###  2.4 Green’s operators

Normally hyperbolic operators are the Lorentzian equivalent of Laplace-type
operators. They carry information about the geometry of the manifold. Green’s
operators are something like inverses for these. The standard example for a
normally hyperbolic operator on $M$ is the d’Alembertian. A canonical analogue
for a normally hyperbolic operator should be available in the noncommutative
setting as well, given as the square of the Dirac operator.

######  Definition 2.4.1.

A normally hyperbolic operator on $M$ is a second order differential operator
on a vector bundle $E$ over $M$ whose principal symbol is given by minus the
metric $g$ (times the identity on fibres).

######  Fixed Notation 2.4.2.

For the rest of this thesis, let $P$ be a normally hyperbolic operator on a
vector bundle $E$ over $M$.

######  Remark 2.4.3.

As $M$ and $P$ are arbitrary, theorems proved for $M$ or $P$ will hold for
arbitrary globally hyperbolic manifolds and arbitrary normally hyperbolic
operators. This will be exploited occasionally. Note that the restriction of a
normally hyperbolic operator to an open subset is again normally hyperbolic.

We shall occasionally need the connection on $E$ induced by $P$ (see [BGP07,
Lemma 1.5.5]):

######  Proposition 2.4.4.

There is a unique connection $\nabla$ on $E$ and a bundle endomorphism $B$
such that

$P=\square^{\nabla}+B,$

where $\square^{\nabla}$ denotes the connection d’Alembert operator associated
to $\nabla$.

######  Fixed Notation 2.4.5.

From now on, let $\nabla$ denote the connection described above.

The definition of the connection d’Alembertian may be found in [BGP07, Example
1.5.2], but is not relevant for our purpose. All we need about $\nabla$ is the
following Leibniz rule (see [BGP07, Lemma 1.5.6]):

######  Proposition 2.4.6.

For $f\in C^{\infty}(M)$ and $g\in\Gamma(E)$, we have

$P(fg)=fPg-2\nabla_{grad(f)}g+(\square f)g,$

where $\square=-\operatorname{div}\operatorname{grad}$ denotes the standard
d’Alembertian.

On the usual spaces of functions, $P$ will not be invertible. Instead, we have
a well-posed Cauchy problem (see [BGP07, Theorem 3.2.11]). This means, to get
something like invertibility, we need to fix “initial conditions”. Fixing
these to be zero at “time $\pm\infty$”, we obtain the advanced and retarded
Green’s operators (see [BGP07, Definition 3.4.1 and Corollary 3.4.3]):

######  Proposition 2.4.7.

There are unique operators

$G^{\pm}\colon\Gamma_{c}(E)\rightarrow\Gamma(E)$

such that for any $\phi\in\Gamma_{c}y(E)$:

  1. 1.

$PG^{\pm}\phi=\phi$

  2. 2.

$G^{\pm}P\phi=\phi$

  3. 3.

$\operatorname{supp}(G^{\pm}\phi)\subseteq J_{\pm}(\operatorname{supp}(\phi))$

######  Definition 2.4.8.

The Operators $G^{\pm}$ are called the advanced/retarded Green’s operator for
$P$. They will be denoted as above, or, if it is not clear which normally
hyperbolic operator is referred to, by $G_{P}^{\pm}$.

The Green’s operators can be extended to other function spaces. The extension
sketched here is described in more detail in [Bär14]. First, we note that the
value $G^{\pm}\phi(x)$ only depends on the value of $\phi$ in the past/future
of $x$, as a change in $\phi$ only affects the future/past of the difference.
Thus, rather than requiring $\operatorname{supp}(\phi)$ to be compact, it
suffices to require that $\operatorname{supp}(\phi)\cap J_{\mp}(x)$ is compact
in order to define $G^{\pm}\phi(x)$: We can multiply by a cut-off function
$\chi$ that is 1 around $\operatorname{supp}(\phi)\cap J_{\mp}(x)$.
$G^{\pm}(\chi\phi)(x)$ will not depend on the choice of $\chi$ and can thus be
used as a definition for $G^{\pm}\phi(x)$. That means, if $\phi$ has
past/future compact support, $G^{\pm}\phi(x)$ can be defined for every $x$. In
this way $G^{\pm}$ can be extended to all past/future compactly supported
smooth functions. $G^{\pm}$ can then also be extended to distributions in the
usual way using duality. Overall, we get a map on past/future compactly
supported distributions that still has all the defining properties.

######  Proposition 2.4.9.

$G^{\pm}$ has unique continuous extensions to maps
$\mathcal{D}^{\prime}_{\pm}(E)\rightarrow\mathcal{D}^{\prime}_{\pm}(E)$ and
$\Gamma_{\pm}(E)\rightarrow\Gamma_{\pm}(E)$ (that will be denoted the same
way). Moreover, for any $\phi\in\mathcal{D}^{\prime}_{\pm}(E)$, we still have

  1. 1.

$PG^{\pm}\phi=\phi$

  2. 2.

$G^{\pm}P\phi=\phi$

  3. 3.

$\operatorname{supp}(G^{\pm}\phi)\subseteq J_{\pm}(\operatorname{supp}(\phi))$

###### Proof.

See [Bär14, Theorem 3.8, Corollary 3.11 and Lemma 4.1] ∎

######  Remark 2.4.10.

As a consequence of the above, we can view $G_{P}^{\pm}$ as an actual inverse
to $P$, by considering both as operators
$\Gamma_{\pm}(M)\rightarrow\Gamma_{\pm}(M)$.

For suitable subsets, restricting the Green’s operator on $M$ gives the
Green’s operator on the subset. Global hyperbolicity ensures that the subset
has a unique Green’s operator, while causal compatibility ensures that there
is no influence from the remainder of $M$.

######  Proposition 2.4.11.

Let $U\subseteq M$ be causally compatible and globally hyperbolic. Then for
$\phi\in\mathcal{D}^{\prime}_{\pm}(E)$, we have

$G^{\pm}_{P|_{U}}(\phi|_{U})=(G^{\pm}_{P}\phi)|_{U}$

###### Proof.

For compactly supported smooth $\phi$, this follows from [BGP07, Proposition
3.5.1.]. By continuity, the statement extends to distributions. ∎

######  Definition 2.4.12.

For an operator $T\colon\Gamma_{c}(E)\rightarrow\Gamma(E)$, define the formal
adjoint (if it exists) to be the operator
$T^{*}\colon\Gamma_{c}(E^{*})\rightarrow\Gamma(E^{*})$ satisfying for all
$\phi\in\Gamma_{c}(E^{*})$, $\psi\in\Gamma_{c}(E)$

$\int\limits_{M}\phi T\psi d\operatorname{Vol}=\int\limits_{M}T^{*}\phi\psi
d\operatorname{Vol}.$

######  Remark 2.4.13.

Note that taking the adjoint is linear, not antilinear as the Hilbert space
adjoint. The reason for this is that for Hilbert spaces the identification
with the dual is antilinear. Beware that the adjoint as defined here does not
coincide with the Hilbert space adjoint for scalar valued functions when
linearly identyfiyng ${\mathbb{C}}^{*}$ with ${\mathbb{C}}$.

The formal adjoint of a normally hyperbolic operator is again normally
hyperbolic (as partial integration doesn’t change the principal symbol of a
second order differential operator).

The dual of the Green’s operators are the Green’s operators of the dual
operator. However, the advanced/retarded sign gets flipped in the process. As
one might expect, this also works when taking powers.

######  Proposition 2.4.14.

For $m\in{\mathbb{N}}$, $(G^{\mp}_{P^{*}})^{m}$ is a formal adjoint for
$(G^{\pm}_{P})^{m}$.

###### Proof.

Let $\phi\in\Gamma_{c}(E)$ and $\psi\in\Gamma_{c}(E^{*})$. Choose $\chi\in
C_{c}^{\infty}(M)$ that is constantly $1$ around
$J_{\pm}(\operatorname{supp}(\phi))\cap J_{\mp}(\operatorname{supp}(\psi))$.
Then $\chi$ is $1$ around

$\operatorname{supp}((G^{\pm}_{P})^{m}\phi)\cap\operatorname{supp}((G^{\mp}_{P^{*}})^{m}\psi)$

and thus we can calculate

$\displaystyle\int\limits_{M}\psi(x)((G^{\pm}_{P})^{m}\phi(x))dx$
$\displaystyle=\int\limits_{M}(P^{*})^{m}(G^{\mp}_{P^{*}})^{m}\psi(x)((G^{\pm}_{P})^{m}\phi(x))dx$
$\displaystyle=\int\limits_{M}(P^{*})^{m}\chi(G^{\mp}_{P^{*}})^{m}\psi(x)(\chi(G^{\pm}_{P})^{m}\phi(x))dx$
$\displaystyle=\int\limits_{M}\chi(G^{\mp}_{P^{*}})^{m}\psi(x)(P^{m}\chi(G^{\pm}_{P})^{m}\phi(x))dx$
$\displaystyle=\int\limits_{M}(G^{\mp}_{P^{*}})^{m}\psi(x)(P^{m}(G^{\pm}_{P})^{m}\phi(x))dx$
$\displaystyle=\int\limits_{M}(G^{\mp}_{P^{*}})^{m}\psi(x)(\phi(x))dx.\qed$

###  2.5 Riesz distributions and Hadamard coefficients

In this section, we introduce Riesz distributions and Hadamard coefficients,
which will play a central role in this thesis. We will follow [BGP07] here,
they are also described in [Gün88] and [Fri75]. As all constructions rely
heavily on the exponential map, the objects will only be defined on convex
subsets of $M$. Note that their values will be dependent on the choice of
convex subset.

######  Fixed Notation 2.5.1.

For this section fix a geodesically convex open subset $U$ of $M$.

The Riesz distributions on $U$ are roughly speaking a Lorentzian analogue to
”powers of the distance functions”. They are described in detail in [BGP07,
sections 1.2 and 1.4]. In this thesis, they will play a role comparable to
that of powers of $x$ in a Taylor series. Before we can define Riesz
distributions on subsets of $M$, we will first define them on Lorentzian
vector spaces (in particular each tangent space of $M$), where they are
roughly speaking ”powers of the norm”.

######  Definition 2.5.2.

Let $V$ be a Lorentzian vector space with Lorentzian bilinear form $\eta$. For
$x\in V$ define

$\gamma^{V}(x):=-\eta(x,x).$

Usually, there will be no ambiguity with respect to what $V$ is and we will
drop it from the notation.

Note that $\gamma(x)$ strictly positive/negative if and only if $x$ is
timelike/spacelike.

######  Definition/Proposition 2.5.3.

Let $V$ be a d-dimensional Lorentzian vector space. For
$\alpha\in{\mathbb{C}}$ with $\Re(\alpha)\geq 0$, define Riesz distributions
$R^{V}_{\pm}(\alpha)$ as the distribution on $V$ given by the function

$R^{V}_{\pm}(\alpha)(x)=\begin{cases}c_{\alpha}\gamma(x)^{\frac{\alpha-d}{2}}&\text{,
if }x\in J_{\pm}(0)\\\ 0&\text{, if }x\notin J_{\pm}(0)\\\ \end{cases}$

with

$c_{\alpha}:=\frac{2^{1-\alpha}\pi^{\frac{2-d}{2}}}{\Gamma(\frac{a}{2})\Gamma(\frac{\alpha-d+2}{2})}.$

The map $\alpha\mapsto R^{V}_{\pm}(\alpha)$ is holomorphic as a map into
$\mathcal{D}^{\prime}(V)$ and extends uniquely to a holomorphic map on all of
${\mathbb{C}}$. For arbitrary $\alpha\in{\mathbb{C}}$, define
$R^{V}_{\pm}(\alpha)$ to be the value of this holomorphic extension. (The
prefactor $c_{\alpha}$ is chosen such that $\square
R^{V}_{\pm}(\alpha+2)=R^{V}_{\pm}(\alpha)$.) In case it is clear what $V$ is,
we omit it from the notation.

######  Remark 2.5.4.

Another way of thinking about the Riesz distribution $R^{\pm}(2k)$ is as
(advanced or retarded) fundamental solutions to $\square^{k}$ (c.f Proposition
3.2.2). In this sense, $R^{\pm}(2)$ can be interpreted as Lorentzian versions
of the Newtonian potential.

###### Proof.

See [BGP07, Lemma 1.2.2] and note that everything remains valid under
isometries, so we may replace Minkowski space with an arbitrary Lorentzian
vector space. ∎

We transport this from tangent spaces to our convex subset $U$ using the
exponential map:

######  Definition 2.5.5.

For $x,y\in U$ and $\alpha\in{\mathbb{C}}$, define the ”squared geodesic
distance”

$\Gamma^{U}_{x}(y):=\gamma\left((exp_{x}^{U})^{-1}(y)\right)$

and define the Riesz distributions

$R^{U}_{\pm}(\alpha,x):=(exp^{U}_{x})^{-1*}(R^{T_{x}M}_{\pm}(\alpha)|_{Dom(exp^{U}_{x})})$

on $U$. We may drop the choice of neighborhood $U$ from the notation if it is
clear from the context (in this section it will always be the one we fixed).

Note that $\Gamma^{U}_{x}$ is positive precisely on $J^{U}(x)$ and
$R^{U}_{\pm}(\alpha,x)$ is supported in $J^{U}_{\pm}(x)$ by Proposition
2.2.20.

######  Remark 2.5.6.

When identifying a Lorentzian vector space $V$ with its tangent space at $0$
in the canonical way, the exponential map at $0$ becomes the identity and thus
$\Gamma_{0}^{V}=\gamma^{V}$ and $R^{V}_{\pm}(\alpha,0)=R^{V}_{\pm}(\alpha)$.

The Riesz distributions have the following properties (see [BGP07, Proposition
1.4.2]222for 4, use that both sides are continuous to also obtain the
statement for $\alpha=0$)

######  Theorem 2.5.7.

For $x\in U$ and $\alpha\in{\mathbb{C}}$, we have the following

  1. 1.

$\alpha\mapsto R_{\pm}(\alpha,x)$ is holomorphic as a distribution-valued map.

  2. 2.

$\operatorname{supp}(R_{\pm}(\alpha,x))\subseteq J^{U}_{\pm}(x)$

  3. 3.

For $\Re(\alpha)\geq d$, $R_{\pm}(\alpha,x)$ is given by

$R_{\pm}(\alpha,x)(y)=\begin{cases}c_{\alpha}\Gamma_{x}(y)^{\frac{\alpha-d}{2}}&\text{,
if }y\in J^{U}_{\pm}(x)\\\ 0&\text{, if }y\notin J^{U}_{\pm}(x)\\\
\end{cases}.$

In particular, for any $k\in{\mathbb{N}}$, $R_{\pm}(\alpha,x)$ is $C^{k}$ for
any $\Re(\alpha)\geq d+2k$.

  4. 4.

$2\alpha\square
R_{\pm}(\alpha+2,x)=\left(\square\Gamma_{x}-2d+2\alpha\right)R_{\pm}(\alpha,x)$

  5. 5.

$2\alpha\operatorname{grad}{R_{\pm}(\alpha+2,x)}=(\operatorname{grad}\Gamma_{x})R_{\pm}(\alpha,x)$

  6. 6.

$R_{\pm}(0,x)=\delta_{x}$

We now turn to the second class of objects we want to define in this section:
the Hadamard coefficients. The motivation for defining the Hadamard
coefficients is that we want to have the formal equality

$P\sum\limits_{k=0}^{\infty}V^{k}(x)R_{\pm}(2k+2,x)=\delta_{x}$

for some smooth sections $V^{k}(x)$. This requires us to investigate the
application of $P$ to a product involving Riesz distributions. To keep the
formulas shorter, we introduce the following differential operator that we’ll
use to describe the remainder term arising in the following.

######  Definition 2.5.8.

Define for $V\in\Gamma(E\otimes E_{x})$:

$\rho_{x}^{U}V:=\nabla_{\operatorname{grad}\Gamma_{x}^{U}}V-\left(\frac{1}{2}\square\Gamma^{U}_{x}-d\right)V.$

The reader may immediately forget the definition of $\rho_{x}^{U}$ and only
remember the following property:

######  Proposition 2.5.9.

We have for $\alpha\in C$, $x\in U$ and $V\in\Gamma(E|_{U}\otimes E^{*}_{x})$

$\alpha
P(VR_{\pm}(\alpha+2,x))=\alpha(PV)R_{\pm}(\alpha+2,x)-((\rho_{x}^{U}-\alpha)V)R_{\pm}(\alpha,x)$

###### Proof.

We calculate for $\Re(\alpha)>d+4$ (so $R_{\pm}(\alpha,x)$ is $C^{2}$ and we
don’t have to worry about distributions)

$\displaystyle\alpha P(VR_{\pm}(\alpha+2,x))$
$\displaystyle=\alpha\left((PV)R_{\pm}(\alpha+2,x)-2\nabla_{\operatorname{grad}R_{\pm}(\alpha+2,x)}V+V\square
R_{\pm}(\alpha+2,x)\right)$
$\displaystyle=\alpha(PV)R_{\pm}(\alpha+2,x)-(\nabla_{\operatorname{grad}\Gamma}V)R_{\pm}(\alpha,x)+\frac{1}{2}V\left(\square\Gamma_{x}-2d+2\alpha\right)R_{\pm}(\alpha,x)$
$\displaystyle=\alpha(PV)R_{\pm}(\alpha+2,x)+\left((\frac{1}{2}\square\Gamma-d+\alpha)V-\nabla_{\operatorname{grad}\Gamma}V\right)R_{\pm}(\alpha,x)$
$\displaystyle=\alpha(PV)R_{\pm}(\alpha+2,x)-((\rho_{x}^{U}-\alpha)V)R_{\pm}(\alpha,x).$

As both sides are holomorphic, the equation holds for arbitrary $\alpha$. ∎

We are now ready to define the Hadamard coefficients (see [BGP07, Definition
2.2.1 and Proposition 2.3.1]):

######  Definition/Proposition 2.5.10.

For every $x\in U$ there is a unique family of sections
$V^{k,U}_{x}\in\Gamma(E\otimes E^{*}_{x}|_{U})$ (indexed by
$k\in{\mathbb{N}}$) that satisfies the transport equations

$(\rho_{x}^{U}-2k)V^{k,U}_{x}=2kPV^{k-1,U}_{x}$

(for $k=0$, the right hand side is set to $0$) subject to the initial
condition

$V^{0,U}_{x}(x)=1.$

These sections are called the Hadamard coefficients (for $P$). The map
$(y,x)\mapsto V^{k,U}_{x}(y)$ is a smooth section in $E\boxtimes
E^{*}|_{U\times U}$. As before, we will drop the superscript where it is not
needed and write $V^{k}_{x}$ for $V^{k,U}_{x}$.

These are defined so that we have the equality:

$P\sum\limits_{k=0}^{N}V^{k}_{x}R_{\pm}(2k+2,x)=\delta_{x}+(PV^{N}_{x})R_{\pm}(2N+2,x).$

Morally we want to replace $N$ by $\infty$. In that case, we would (formally)
obtain a solution to the equation

$PF=\delta,$

which (together with support conditions) determines the Schwartz kernels of
the Green’s operators.

However, in general the remainder $(PV^{N}_{x})R_{\pm}(2N+2,x)$ may not vanish
for $N\rightarrow\infty$. All we get is that the remainder becomes arbitrarily
differentiable for $N$ large enough, i.e. we get an asymptotic expansion in
differentiability order. This is the motivation for the definition of the
Hadamard coefficients and one of the key results that will be used in this
thesis: We have an asymptotic expansion

$G^{\pm}\delta_{x}|_{U}\sim_{x}\sum\limits_{k=0}^{\infty}V^{k,U}_{x}R^{U}_{\pm}(2k+2,x)$

in case $U$ satisfies some further assumptions. This expansion is shown in
[BGP07, Proposition 2.5.1] on some suitable small neighborhoods and we will
show a version for slightly different neighborhoods later on.

######  Remark 2.5.11.

Note that both $G^{\pm}\delta_{x}$ and $R^{U}_{\pm}(2k+2,x)$ are supported
inside $J_{\pm}(x)$ hence the same is true for the remainder term in the
expansion. Thus all derivatives that the remainder term has must be vanishing
at the boundary of $J_{\pm}(x)$, in particular at $x$. This means that
differentiability of the remainder leads to decay near $x$ of the
corresponding order (see [BGP07, Theorem 2.5.2] for a detailed statement).

There is another propagator associated to $P$, known as the Feynman
propagator, that also has a similar asymptotic expansion involving the
Hadamard coeficients. However, in the Feynman case nothing needs to vanish
anywhere, so differentiability will not transform into decay estimates. That
is the reason why we need the causal propagator in the approach presented
here.

In general, as the transport equations depend on $\Gamma^{U}$, the value
$V^{k,U}_{x}(y)$ will depend on the choice of neighborhood $U$. This is due to
the fact that, in different convex neighborhoods, the unique geodesic joining
$x$ and $y$ might be different. If, however, one of the two sets is contained
within the other, their coefficients coincide:

######  Proposition 2.5.12.

Let $U$ and $U^{\prime}$ be open convex subsets of $M$ with
$U^{\prime}\subseteq U$. Then for any $x\in U^{\prime}$,

$V^{k,U}_{x}|_{U^{\prime}}=V^{k,U^{\prime}}_{x}$

and

$R^{U}_{\pm}(\alpha,x)|_{U^{\prime}}=R^{U^{\prime}}_{\pm}(\alpha,x).$

###### Proof.

Any geodesic in $U^{\prime}$ is also in $U$, so for $x$ and $y$ in
$U^{\prime}$ the unique geodesics joining them in $U$ and $U^{\prime}$ are the
same. This means that $exp_{x}^{-1}(y)$ and hence $\Gamma_{x}(y)$ coincide for
$U$ and $U^{\prime}$. Thus their transport equations are the same, so
$V^{k,U}_{x}|_{U^{\prime}}$ also solves the transport equations for
$U^{\prime}$.

Moreover, as $\Gamma$ coincides for both sets, the same is true for
$R_{\pm}(\alpha,x)$ if $\Re(\alpha)$ is large enough. By uniqueness of
holomorphic continuation, the Riesz distributions also coincide for arbitrary
$\alpha$. ∎

######  Corollary 2.5.13.

For any two convex open neighborhoods $U_{1}$ and $U_{2}$ of $p\in M$, there
is a neighborhood $U_{0}\subseteq U_{1}\cap U_{2}$ of $p$ such that for any
$k\in{\mathbb{N}}$

$V^{k,U_{1}}_{x}|_{U_{0}}=V^{k,U_{2}}_{x}|_{U_{0}}.$

###### Proof.

Take $U_{0}$ to be any open convex neighborhood contained in $U_{1}\cap
U_{2}$. Then both sides of the equation are equal to $V^{k,U_{0}}_{x}$. ∎

Thus the germ of the Hadamard coefficients at the diagonal (i.e. at $y=x$) is
independent of the choice of neighborhood, allowing us to define:

######  Definition 2.5.14.

For any local operator $L$, write $LV^{k}_{x}(x)$ for the value of
$LV^{k,U}_{x}(x)$ for any (and hence all) convex open neighborhood(s) $U$ of
$x$.

###  2.6 Evolution operators

For this section, we will assume that $M$ is foliated into smooth spacelike
Cauchy hypersurfaces $(\Sigma_{t})_{t\in{\mathbb{R}}}$. By Theorem 2.2.12,
such a foliation always exists for globally hyperbolic manifolds. Let $n$
denote the future directed unit normal vector field to the hypersurfaces
$\Sigma_{t}$.

Given initial data on any $\Sigma_{t}$, there is a unique solution to $Pu=0$
with these initial data (see [BGP07, Theorem 3.2.11]):

######  Theorem 2.6.1 (Well-posedness of the Cauchy problem).

For any $t\in{\mathbb{R}}$ and $f,g\in\Gamma_{c}(E|_{\Sigma_{t}})$, there is a
unique $u\in\Gamma(E)$ such that

$u|_{\Sigma_{t}}=f,$ $\nabla_{n}u|_{\Sigma_{t}}=g$

and

$Pu=0.$

This $u$ is supported in
$J(\operatorname{supp}(f)\cup\operatorname{supp}(g))$.

This allows us to define the following family of evolution operators:

######  Definition 2.6.2.

For $s,t\in{\mathbb{R}}$, define

$Q(t,s)\colon\Gamma_{c}(E|_{\Sigma_{s}})\rightarrow\Gamma_{c}(E|_{\Sigma_{t}})$

by setting for $\psi\in\Gamma_{c}(E|_{\Sigma_{s}})$

$Q(t,s)\psi(x):=u(x,t)$

for the unique $u\in\Gamma(E)$ with

$u|_{\Sigma_{s}}=0$ $\nabla_{n}u|_{\Sigma_{s}}=\psi$

and

$Pu=0.$

######  Remark 2.6.3.

The full Cauchy evolution operator would be the operator

$\displaystyle\Gamma_{c}(E|\sigma_{s})\oplus\Gamma_{c}(E|\sigma_{s})$
$\displaystyle\rightarrow\Gamma_{c}(E|\sigma_{t})\oplus\Gamma_{c}(E|\sigma_{t})$
$\displaystyle(u|_{\Sigma_{s}},\nabla_{n}u|_{\Sigma_{s}})$
$\displaystyle\mapsto(u|_{\Sigma_{t}},\nabla_{n}u|_{\Sigma_{t}})\text{ for
}Pu=0.$

The operator we consider here is the top right entry of the full operator
viewed as a $2\times 2$-matrix.

We want to relate this system of evolution operators to the Green’s operators
of $P$. For that we need a slightly changed version of [BGP07, Lemma 3.2.2].
The original version in [BGP07] phrased in terms of fundamental solutions
rather than Green’s operators and only for suitable small neighborhoods, as
[BGP07] has only shown existence of fundamental solutions (which are basically
kernels of Green’s operators) on these neighborhoods at that point. However,
as we have already taken the global existence of Green’s operators for
granted, we may state the lemma for all of $M$.

######  Lemma 2.6.4.

For $\psi\in\Gamma(E|_{\Sigma_{s}})$, let $u$ be a solution to

$u|_{\Sigma_{s}}=0$ $\nabla_{n}u|_{\Sigma_{s}}=\psi$

and

$Pu=0.$

Then we have for every $\phi\in\Gamma_{c}(E^{*})$:

$\int\limits_{M}\phi(x)u(x)dx=-\int\limits_{\Sigma_{s}}G_{P^{*}}\phi(x)\psi(x)dx.$

###### Proof.

In the proof of [BGP07, Lemma3.2.2], set $S:=\Sigma_{s}$, $u_{1}:=\psi$ and
$u_{0}:=0$, replace $\Omega$ with $M$ and redefine $\Psi$ as
$G^{+}_{P^{*}}(\phi)$ and $\Psi^{\prime}$ as $G^{-}_{P^{*}}(\phi)$. The proof
then shows the claimed result. ∎

With this we can show the following:

######  Proposition 2.6.5.

For any $\psi\in\Gamma(E|_{\Sigma_{s}})$, we have

$Q(t,s)\psi=G(\iota_{s*}\psi)|_{\Sigma_{t}},$

where $\iota_{s}$ denotes the inclusion of $\Sigma_{s}$ into $M$.

###### Proof.

Let $u$ be as above. Then by definition

$Q(t,s)\psi=u|_{\Sigma_{t}}.$

Recall that the Green’s operators of $P^{*}$ are adjoints to that of $P$ with
flipped $\pm$ (see Proposition 2.4.14), so $G^{*}=-G_{P^{*}}$. Thus, using
Lemma 2.6.4, we get for any $\phi\in\Gamma_{c}(E^{*})$:

$u[\phi]=-\int\limits_{\Sigma_{s}}G_{P^{*}}\phi(x)\psi(x)dx=\iota_{s*}\psi[-G_{P^{*}}\phi]=G(\iota_{s*}\psi)[\phi].$

Thus

$u=G(\iota_{s*}\psi)$

and hence

$Q(t,s)\psi=u|_{\Sigma_{t}}=G(\iota_{s*}\psi)|_{\Sigma_{t}}.\qed$

Morally, this means that

$\mathcal{K}(Q(t,s))(y,x)=\mathcal{K}(G)(y,x)$

for any $x\in\Sigma_{s}$ and $y\in\Sigma_{t}$.

###  2.7 Wavefront calculus

####  2.7.1 Wavefront calculus on ${\mathbb{R}}^{n}$

The wavefront set of a distribution $\eta$ is a refinement of its singular
support. It captures not only at which points a distribution is not smooth,
but also in which directions. It will be a subset of the cotangent bundle of
the manifold without the zero section. The notion of wavefront sets goes back
to Lars Hörmander ([Hör71]).

######  Definition 2.7.1.

For a manifold $X$, let $\dot{T}^{*}(X)$ denote the set of all non-zero
vectors in the cotangent bundle $T^{*}(X)$. In this part, we will assume that
$X$ and $Y$ are open subset of ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$, for
$m,n\in{\mathbb{N}}$. In this case, we identify $\dot{T}^{*}(X)$ with
$X\times({\mathbb{R}}^{n}\backslash\\{0\\})$. Compactly supported functions on
$X$ are sometimes implicitly extended by $0$ to functions on all of
${\mathbb{R}}^{n}$. All definitions made in terms of $X$ and $Y$ should be
understood as definitions for arbitrary manifolds, if this is possible.

We define the wavefront of a distribution by characterizing its complement:

######  Definition 2.7.2.

Let $\eta$ be a distribution on ${\mathbb{R}}^{n}$. A pair
$(x,\xi)\in\dot{T}^{*}(X)$ is not in the wavefront set
$\operatorname{WF}(\eta)\subseteq\dot{T}^{*}(X)$, if and only if the following
is satisfied:

There is a conic neighborhood $O$ of $\xi$ and $\chi\in C_{c}^{\infty}(X)$
such that

  * •

$x\in\operatorname{supp}(\chi)$

  * •

the Fourier transform of $\chi\eta$ decays rapidly on $O$, i.e. we have for
all $k\in{\mathbb{N}}$:

$\sup\limits_{\xi^{\prime}\in
O}(1+|\xi^{\prime}|)^{k}\|\mathcal{F}(\chi\eta)(\xi^{\prime})\|<\infty.$

Here conic means that $\lambda\xi^{\prime}\in O$ for $\xi^{\prime}\in O$ and
$\lambda\in(0,\infty)$.

As multiplication with $\xi$ in Fourier space corresponds to differentiation
in the corresponding direction, the wavefront set can be interpreted as the
set of directions at each point in which the function is not smooth.

The space of all distributions with a given wavefront set can then be
topologized by taking something like the best constants in the above condition
as additional seminorms:

######  Definition 2.7.3.

For a distribution $\eta$ on $X$, $\chi\in C_{c}^{\infty}(X)$,
$V\subseteq{\mathbb{R}}^{n}\backslash\\{0\\}$ closed and conic and
$k\in{\mathbb{N}}$, define

$\|\eta\|_{k,V,\chi}:=\sup\limits_{\xi\in
V}(|\xi|+1)^{k}|\mathcal{F}(\chi\eta)(\xi)|.$

For $\Lambda\subseteq\dot{T}^{*}({\mathbb{R}}^{n})$ a closed conic subset (in
the sense that $(x,\mu\xi)\in\Lambda$ for $(x,\xi)\in\Lambda$ and
$\mu\in(0,\infty)$), let $\mathcal{D}^{\prime}_{\Lambda}(X)$ be the space of
all distributions on $X$ with wavefront contained in $\Lambda$. This is
topologized by the seminorms for the (strong) topology on
$\mathcal{D}^{\prime}(X)$ and additional seminorms

$\|\cdot\|_{k,V,\chi}$

for any $k,V,\chi$ as above such that

$\operatorname{supp}(\chi)\times V\cap\Lambda=\emptyset.$

The use of the strong topology on $\mathcal{D}^{\prime}$ is required to make
some operations continuous on these spaces. Moreover, with this topology,
$D^{\prime}_{\Lambda}$ is complete (see [DB13, Corollary 25]). However, for
convergence of sequences it is equivalent to the weak*-topology. As sequential
continuity would be sufficient for the purpose of this paper, we could also
work with the weak*-topology instead.

As for the usual space of distributions, $C_{c}^{\infty}(X)$ is dense in
$\mathcal{D}^{\prime}_{\Lambda}(X)$ for any $\Lambda$ (see [Hör03, 8.23]).
Obviously, for $\Theta\subseteq\Lambda$, $\mathcal{D}^{\prime}_{\Theta}(X)$ is
continuously embedded in $\mathcal{D}^{\prime}_{\Lambda}(X)$, as any seminorm
of the latter is also a seminorm of the former.

The wavefront set is a refinement of the singular support, in the sense that
$\operatorname{singsupp}(\eta)$ is the projection of $\operatorname{WF}(\eta)$
onto the first component. Moreover, $\mathcal{D}^{\prime}_{\emptyset}(X)$ and
$C^{\infty}(X)$ coincide as topological vector spaces (see [BDH16, Lemma
7.2]).

To discuss continuity properties of certain bilinear operations on wavefront
spaces, we shall need the concept of hypocontinuity:

######  Definition 2.7.4.

A bilinear map $b\colon V_{1}\times V_{2}\rightarrow V_{3}$, for $V_{i}$
topological vector spaces, is called hypocontinuous, if for every neighborhood
$W$ of 0 in $V_{3}$ and any bounded sets $A_{1}\subset V_{1}$ and
$A_{2}\subset V_{2}$ there are neighborhoods of zero $U_{1}\subseteq V_{1}$
and $U_{2}\subseteq V_{2}$ such that $b^{-1}(W)$ contains $U_{1}\times A_{2}$
and $A_{1}\times U_{2}$.

If the topologies are given in terms of seminorms, this can be rephrased in
terms of those as follows: For every bounded subset $A\subset V_{1}$ and every
seminorm $\rho$ of $V_{3}$, there are seminorms $(\rho_{i})_{i\leq N}$ of
$V_{2}$ and $C_{A}\in{\mathbb{R}}$ such that for all $v\in A$, $w\in V_{2}$,
we have

$\rho(b(v,w))\leq C_{A}\sum\limits_{i=0}^{N}\rho_{i}(w)$

and the same holds with $V_{1}$ and $V_{2}$ interchanged.

As any bounded set is contained in a suitably scaled version of every
neighborhood of zero, this is weaker than continuity. As each bounded set
$A_{i}$ is an open subset of itself and translations of bounded sets are
bounded, a hypocontinuous map is continuous when restricted to $A_{1}\times
V_{2}$ or $V_{1}\times A_{2}$. In particular, as points are bounded, it is
separately continuous in each argument when keeping the other argument fixed.
Moreover, since all convergent sequences are bounded, hypocontinuous maps are
sequentially continuous. As differential quotients can be defined via limits
of sequences, hypocontinuous maps preserve differentiability and
holomorphicity. As images of bounded sets under continuous linear maps are
bounded and preimages of open sets are open, compositions of hypocontinuous
bilinear maps with continuous linear maps (either postcomposition or
precomposition in either argument) are again hypocontinuous. The bilinear
operations we will encounter in wavefront calculus are generally not
continuous, but hypocontinuous.

One of the advantages of wavefront calculus is that it allows us to perform
operations that are not well-defined for general distributions, like
multiplications or pull-backs, on distributions with the right wavefront. The
principle is always (mostly) the same: If the image of some closed conic
source wavefront set(s) is contained in a target wavefront set, then the
operation is well-defined and continuous between the associated wavefront
spaces. In particular, the operation is well-defined if the image of the
wavefront set(s) does not intersect the zero section. We now define the images
of wavefront sets under the operations that will be needed later:

######  Definition 2.7.5.

For a smooth map $f\in C^{\infty}(X,Y)$ and closed conic sets
$\Lambda,\Lambda^{\prime}\subseteq\dot{T}^{*}(X)$ and
$\Theta\subseteq\dot{T}^{*}(Y)$, define

  * •

$f^{*}(\Theta):=\\{(df_{x})^{*}(\xi)\mid x\in X,\xi\in\Theta\cap
T_{f(x)}(Y)\\}$

  * •

$f_{*}(\Lambda):=\\{\xi\in\dot{T}^{*}(Y)\mid\exists x\in X\colon\xi\in
T_{f(x)}(Y)\wedge(df_{x})^{*}\xi\in\Lambda\cup 0\\}$

  * •

$\Lambda\bar{+}\Lambda^{\prime}:=\\{\xi+\xi^{\prime}\mid\xi\in\Lambda,\xi^{\prime}\in\Lambda^{\prime}\\}\cup\Lambda\cup\Lambda^{\prime}$

  * •

$\Lambda\bar{\times}\Theta:=((\Lambda\cup 0)\times(\Theta\cup 0))\backslash 0$

(Here $(df_{x})^{*}$ denotes the adjoint of the total differential of $f$ at
$x$).

In the case of pushforwards, there is an additional restriction that is
required even in the case of smooth functions:

$f_{*}(\phi)[\psi]:=\phi[\psi\circ f]$

is not well defined in general, as $\psi\circ f$ may not be compactly
supported. We thus have to restrict to those distributions with a suitable
support.

######  Definition 2.7.6.

For $C\subseteq X$ and $\Lambda\subseteq\dot{T}^{*}(X)$, let
$\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}$ denote the subspace of
$\mathcal{D}^{\prime}_{\Lambda}(X)$ consisting of those elements with support
in $C$. Let $\Lambda|_{C}$ denote the set of all elements of $\Lambda$ whose
basepoint lies in $C$.

######  Theorem 2.7.7 (wavefront calculus on ${\mathbb{R}}^{n}$).

Let $X$ and $Y$ be open subsets of ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$,
with $n,m\in{\mathbb{N}}$. Let $\Lambda,\Lambda^{\prime}\subseteq\dot{T}^{*}X$
and $\Theta\subseteq\dot{T}^{*}(Y)$ be closed conic subsets. Let $f\colon
X\rightarrow Y$ be a smooth map.

  * •

If $f^{*}(\Theta)$ does not intersect the zero section, $f^{*}$ extends to a
continuous map from $\mathcal{D}^{\prime}_{\Theta}(Y)$ to
$\mathcal{D}^{\prime}_{f^{*}(\Theta)}(X)$.

  * •

If $C\subseteq X$ is a closed set such that $f|_{C}$ is proper, $f_{*}$
defines a continuous map from $\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}$ to
$\mathcal{D}^{\prime}_{f_{*}(\Lambda)}(Y)$.

  * •

If $\Lambda\bar{+}\Lambda^{\prime}$ does not intersect the zero section,
multiplication of functions extends to a hypocontinuous bilinear map

$\mathcal{D}^{\prime}_{\Lambda}(X)\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(X)\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{+}\Lambda^{\prime}}(X).$

  * •

The tensor product defines a hypocontinuous bilinear map

$\mathcal{D}^{\prime}_{\Lambda}(X)\times\mathcal{D}^{\prime}_{\Theta}(Y)\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{\times}\Theta}(X\times
Y).$

This is shown in [BDH16, Proposition 5.1, Theorem 6.3, Theorem 6.1 and Theorem
4.6 ], with one difference: For their push-forward theorem, [BDH16] assume
that their wavefront set $\Lambda$ only contains covectors with basepoint in
$C$. This implies the version above, if we can show the following Lemma:

######  Lemma 2.7.8.

For $C\subseteq X$ closed, we have

$\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}=\mathcal{D}^{\prime}_{\Lambda|_{C}}(X)|_{C}$

as topological vector spaces.

###### Proof.

Note that
$\Lambda|_{C}=\Lambda\cap(C\times({\mathbb{R}}^{n}\backslash\\{0\\}))$ is the
intersection of two closed conic subsets of $\dot{T}^{*}({\mathbb{R}}^{n})$
and hence closed and conic again. As $\Lambda|_{C}\subseteq\Lambda$, we
automatically have a bounded inclusion

$\mathcal{D}^{\prime}_{\Lambda|_{C}}(X)|_{C}\subseteq\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}.$

Fix $\eta\in\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}$. As every element of
$\operatorname{WF}(\eta)$ has basepoint in

$\operatorname{singsupp}(\eta)\subseteq\operatorname{supp}(\eta)\subseteq C,$

$\eta$ has wavefront in $\Lambda|_{C}$, so the two spaces agree as sets. It
remains to show that they have the same topology. Consider some seminorm
$\|\cdot\|_{k,V,\chi}$ of $\mathcal{D}^{\prime}_{\Lambda|_{C}}(X)|_{C}$. Then
$\operatorname{supp}(\chi)\times V$ does not intersect $\Lambda|_{C}$ and
hence $(\operatorname{supp}(\chi)\cap C)\times V$ does not intersect
$\Lambda$. Let $V_{1}$ denote the set of unit vectors in $V$. As $\Lambda$ is
closed, every

$\xi\in(\operatorname{supp}(\chi)\cap C)\times V_{1}$

has an open neighborhood $U(\xi)$ that is disjoint from $\Lambda$. As product
neighborhoods form a basis, we may assume that this is of the form

$U(\xi)=U_{1}(\xi)\times U_{2}(\xi).$

As $V_{1}$ is compact, for each $x\in(\operatorname{supp}(\chi)\cap C)$ there
are finitely many $(v_{i})_{i\in I}\in V_{1}$ such that the corresponding
neighborhoods $U_{2}(x,v_{i})$ cover $V_{1}$. Set
$U^{\prime}(x):=\bigcap\limits_{i\in I}U_{1}(x,v_{i})$. Then
$U^{\prime}(x)\times V_{1}$ does not intersect $\Lambda$, since any
$U^{\prime}(x)\times U_{2}(x,v_{i})$ doesn’t. As
$(\operatorname{supp}(\chi)\cap C)$ is compact, we may find finitely many
$(x_{j})_{j\in J}\in(\operatorname{supp}(\chi)\cap C)$ such that

$O:=\bigcup\limits_{i\in I}U^{\prime}(x_{i})$

contains $(\operatorname{supp}(\chi)\cap C)$. By construction $O\times V_{1}$
does not intersect $\Lambda$. As $\Lambda$ is conic, the same holds for
$O\times V$. Choose $\psi\in C_{c}^{\infty}(O)$ that is $1$ on
$\operatorname{supp}(\chi)\cap C$. Then for arbitrary
$\eta\in\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}$, we have
$\psi\chi\eta=\chi\eta$ and hence

$\|\eta\|_{k,V,\chi}=\|\eta\|_{k,V,\psi\chi}.$

Moreover, $\operatorname{supp}(\psi\chi)\times V\subseteq O\times V$ does not
intersect $\Lambda$, so the right hand side is a seminorm of
$\mathcal{D}^{\prime}_{\Lambda}(X)|_{C}$. Thus the two spaces have the same
seminorms and hence the same topology.

∎

####  2.7.2 Wavefront calculus on manifolds

We now turn from functions on subsets of ${\mathbb{R}}^{n}$ to sections in
vector bundles. Defining the wavefront set and extending the previous results
for sections can be done in a fairly straightforward way by considering their
components in some coordinate chart:

######  Definition 2.7.9.

By a ”bundle coordinate” on a vector bundle $F$ over a manifold $X$, we shall
mean a smooth, fibrewise linear map $F\rightarrow{\mathbb{C}}$ (these can be
canonically identified with sections of $F^{*}$ or bundle homomorphisms
$F\rightarrow X\times{\mathbb{C}}$ over the identity). Let $A$ be a bundle
coordinate or a bundle homomorphism over the identity. For a distribution
$\eta\in\mathcal{D}^{\prime}(F)$ and any scalar test function $\phi$, define

$(A\eta)[\phi]:=\eta[A^{*}\circ\phi],$

where $A^{*}$ is the fibrewise adjoint of $A$.

We have for any $g\in\Gamma(F)$, test section $\phi$ and
$\eta\in\mathcal{D}^{\prime}(M)$:

$A(g\eta)[\phi]=g\eta[A^{*}\circ\phi]=\eta[x\mapsto((A^{*}\phi(x))(g(x)))]=\eta[x\mapsto(\phi(x)(A(g(x))))]=(A\circ
g)\eta[\phi].$

In particular (set $\eta=1$), we have $Ag=A\circ g$.

With this, we define the wavefront set of a distribution in a vector bundle:

######  Definition 2.7.10.

Let $E$ be a vector bundle over a manifold $X$. For
$\eta\in\mathcal{D}^{\prime}(E)$ define the wavefront set of $\eta$ as the
union of all sets of the form

$\psi^{*}(\operatorname{WF}((\psi^{-1})^{*}(A\eta|_{\operatorname{Dom}(\psi)})))$

for coordinate charts $\psi$ and bundle coordinates $A$ defined on
$\operatorname{Dom}(\psi)$. The space $\mathcal{D}^{\prime}_{\Lambda}(E)$ of
all distributions with wavefront in a given closed conic set $\Lambda$ is
topologized by the (strong) seminorms of $\mathcal{D}^{\prime}(E)$ and all
seminorms

$\|\eta\|_{k,V,\chi,\psi,A}:=\|(\psi^{-1})^{*}(A\eta|_{\operatorname{Dom}(\psi)})\|_{k,V,\chi\circ\psi^{-1}}=\sup\limits_{\xi\in
V}(|\xi|+1)^{k}\big{|}\mathcal{F}\big{(}(\psi^{-1})^{*}(\chi
A\eta)\big{)}(\xi)\big{|},$

where $\psi$ and $A$ are as above, $\chi\in
C_{c}^{\infty}(\operatorname{Dom}(\psi))$ is some cut-off, $k\in{\mathbb{N}}$
and $V$ is a closed cone in ${\mathbb{R}}^{n}$ such that
$\operatorname{supp}(\chi\circ\psi^{-1})\times V$ does not intersect
$(\psi^{-1})^{*}(\Lambda)$. Write $\mathcal{D}^{\prime}_{\Lambda}(X)$ for
$\mathcal{D}^{\prime}_{\Lambda}(X\times{\mathbb{C}})$.

Basically all the results on subsets of ${\mathbb{R}}^{n}$ generalize to this
setting. See the appendix (Appendix: wavefront calculus on manifolds) for the
proof of the following theorem:

######  Theorem 2.7.11 (Wavefront calculus on manifolds).

The results obtained for scalar functions on ${\mathbb{R}}^{n}$ carry over to
sections of vector bundles on manifolds. Assume that $X$ and $Y$ are
manifolds. For $i\in\\{1,2,3\\}$, let $E_{i}$ be a vector bundle over $X$ and
let $F$ be a vector bundle over $Y$. Let
$\Lambda,\Lambda^{\prime}\subseteq\dot{T}^{*}(X)$ and
$\Theta\subseteq\dot{T}^{*}(Y)$ be closed conic subsets.

  1. 1.

The seminorms defined for $\mathcal{D}^{\prime}_{\Lambda}(E_{1})$ are finite.
In case $X$ is a subset of ${\mathbb{R}}^{n}$ and $E_{1}=X\times{\mathbb{C}}$,
the new definition coincides with the earlier one.

  2. 2.

$\mathcal{D}^{\prime}_{\Lambda}(E_{1})$ is complete.

  3. 3.

$\Gamma_{c}(E_{1})$ is dense in $\mathcal{D}^{\prime}_{\Lambda}(E_{1})$.

  4. 4.

$\mathcal{D}^{\prime}_{\emptyset}(E_{1})$ coincides with $\Gamma(E_{1})$ as a
topological vector space.

  5. 5.

Differential operators between sections of $E_{1}$ and $E_{2}$ map
$\mathcal{D}^{\prime}_{\Lambda}(E_{1})$ to
$\mathcal{D}^{\prime}_{\Lambda}(E_{2})$ continuously.

  6. 6.

Let $f\colon X\rightarrow Y$ be a smooth map.

     * •

If $f^{*}(\Theta)$ does not intersect the zero section, $f^{*}$ extends to a
continuous map from $\mathcal{D}^{\prime}_{\Theta}(F)$ to
$\mathcal{D}^{\prime}_{f^{*}(\Theta)}(f^{*}(F))$.

     * •

If $C\subseteq X$ is a closed set such that $f|_{C}$ is proper, $f_{*}$
defines a continuous map from $\mathcal{D}^{\prime}_{\Lambda}(f^{*}F)|_{C}$ to
$\mathcal{D}^{\prime}_{f_{*}(\Lambda)}(F)$.

     * •

If $\Lambda\bar{+}\Lambda^{\prime}$ does not intersect the zero section,
pointwise ”multiplication” of sections in $E_{1}\otimes E_{2}^{*}$ and
$E_{2}\otimes E_{3}$, defined at each point by

$(a\otimes b)\cdot(c\otimes d):=b(c)\cdot(a\otimes d)$

extends to a hypocontinuous bilinear map

$\mathcal{D}^{\prime}_{\Lambda}(E_{1}\otimes
E_{2}^{*})\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(E_{2}\otimes
E_{3})\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{+}\Lambda^{\prime}}(E_{1}\otimes
E_{3}).$

     * •

The tensor product defines a hypocontinuous bilinear map

$\mathcal{D}^{\prime}_{\Lambda}(E_{1})\times
D^{\prime}_{\Theta}(F)\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{\times}\Theta}(E_{1}\boxtimes
F).$

     * •

If $\Lambda\bar{+}\Lambda^{\prime}$ does not intersect the zero section and
$K\subseteq X$ is compact, evaluation of distributions extends to
hypocontinuous bilinear maps

$\mathcal{D}^{\prime}_{\Lambda}(E_{1})\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(E_{1}^{*})|_{K}\rightarrow{\mathbb{C}}$

and

$\mathcal{D}^{\prime}_{\Lambda^{\prime}}(E_{1}^{*})|_{K}\times\mathcal{D}^{\prime}_{\Lambda}(E_{1})\rightarrow{\mathbb{C}}.$

These are symmetric, in the sense that

$\zeta[\eta]=\eta[\zeta].$

######  Remark 2.7.12.

The multiplication map described here corresponds to pointwise composition
when identifying tensor products with spaces of linear maps as in 2.1.1. Using
the other identifications of 2.1.1, this contains the following as special
cases (by setting all non-occuring vector bundles to $X\times{\mathbb{C}}$):

  * •

Products of scalar functions

$\mathcal{D}^{\prime}_{\Lambda}(X)\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(X)\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{+}\Lambda^{\prime}}(X).$

  * •

Scalar multiplication

$\mathcal{D}^{\prime}_{\Lambda}(X)\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(E_{3})\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{+}\Lambda^{\prime}}(E_{3})$

  * •

Pointwise tensor product

$\mathcal{D}^{\prime}_{\Lambda}(E_{1})\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(E_{3})\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{+}\Lambda^{\prime}}(E_{1}\otimes
E_{3}).$

  * •

Pointwise dual pairing

$\mathcal{D}^{\prime}_{\Lambda}(E_{2}^{*})\times\mathcal{D}^{\prime}_{\Lambda^{\prime}}(E_{2})\rightarrow\mathcal{D}^{\prime}_{\Lambda\bar{+}\Lambda^{\prime}}(X).$

The tensor product and pushforward interact in the way one would expect:

######  Proposition 2.7.13.

If $f\colon X\rightarrow Y$ and $g\colon X^{\prime}\rightarrow Y^{\prime}$ are
smooth maps between manifolds, we have for distributions $\eta$ and $\zeta$ in
vector bundles over $X$ and $X^{\prime}$ whose pushforward under $f$
respectively $g$ is well-defined:

$(f_{*}\eta)\otimes(g_{*}\zeta)=(f\times g)_{*}(\eta\otimes\zeta),$

where $f\times g(x,x^{\prime}):=(f(x),g(x^{\prime}))$.

###### Proof.

If $f$ and $g$ are smooth and $\phi$ and $\psi$ are test sections, we have

$(f_{*}\eta)\otimes(g_{*}\zeta)[\phi\otimes\psi]=\eta[\phi\circ
f]\cdot\zeta[\psi\circ g]=\eta\otimes\zeta[(\phi\otimes\psi)\circ(f\times
g)]=(f\times g)_{*}(\eta\otimes\zeta)[\phi\otimes\psi].$

Thus

$(f_{*}\eta)\otimes(g_{*}\zeta)=(f\times g)_{*}(\eta\otimes\zeta).$

As smooth functions are dense, this holds for arbitrary distributions. ∎

##  Chapter 3 Asymptotic expansions related to Green’s operators

In this section, we will develop asymptotic expansions in differentiability
orders for the kernels of powers of $G^{\pm}_{P}$ and that of $G^{\pm}_{P-z}$
for $z\in{\mathbb{C}}$. This is inspired by [DW20], where similar results are
obtained for Feynman inverses. However, the methods used here are different.
While the only things necessary for the proof of our formula for the Hadamard
coefficients are the standard Hadamard expansion and the formula for the
$z$-dependent Hadamard coefficients shown in Proposition 3.2.4, the other
results might be of independent interest.

###  3.1 Powers of Green’s operators

We first derive an expansion for powers of the advanced/retarded Green’s
operators. Before we can start the main proof, we need to do some technical
work to guarantee that the Green’s operators do not decrease differentiability
orders too much (locally). This will be necessary to control the remainder
terms of our asymptotic expansion. The result is basically a consequence of
mapping smooth sections to smooth sections continuously (a better result could
be obtained by investigating $G$ as a Fourier integral operator).

######  Proposition 3.1.1.

Let $A\subset M$ be past/future compact compact and $U\subset M$ be open and
relatively compact. Let $r_{U}$ denote restriction to $U$. Then for every
$n\in{\mathbb{N}}$ there is $m\in{\mathbb{N}}$ such that $r_{U}\circ G^{\pm}$
maps $\Gamma^{m}_{A}(E)$ to $\Gamma^{n}(E|_{U})$ continuously.

###### Proof.

We know (see 2.4.9) that

$G^{\pm}\colon\Gamma_{\pm}(E)\rightarrow\Gamma_{\pm}(E)$

is continuous. Since the topologies of $\Gamma_{A}(M)$ and
$\Gamma_{J_{\pm}(A)}(M)$ coincide with the subspace topology from
$\Gamma_{\pm}$ (as the latter carries the limit topology), we know that the
restriction

$G^{\pm}\colon\Gamma_{A}(E)\rightarrow\Gamma_{J_{\pm}(A)}(E)$

is also continuous. $C^{k}$\- norms on ${\overline{U}}$ are seminorms on the
target space. Thus by the seminorm-characterization of continuity, we know
that for any $n\in{\mathbb{N}}$, there is $c\in{\mathbb{R}}$,
$m\in{\mathbb{N}}$ and $K\subseteq M$ such that for all $\phi\in\Gamma_{A}(E)$

$\|G^{\pm}\phi\|_{C^{n}(\overline{U})}\leq c\|\phi\|_{C^{m}(K)}.$

The right hand side is also a seminorm of $\Gamma^{m}_{A}(E)$. Thus
$r_{U}\circ G^{\pm}|_{\Gamma_{A}(E)}$ extends continuously to a map
$\Gamma^{m}_{A}(E)\rightarrow\Gamma^{n}(E|_{U})$. As $C^{k}$\- convergence
implies distributional convergence, this continuous extension must coincide
with $r_{U}\circ G^{\pm}|_{\Gamma^{m}_{A}(E)}$, as the latter is
distributionally continuous. ∎

As Hadamard coefficients and Green’s kernels are sections in $E\boxtimes
E^{*}$ and we want differentiability in both coordinates, we need to extend
the above to the case where $G^{\pm}$ acts in the first component on sections
of a box product.

######  Proposition 3.1.2.

Let $A\subset M$ be past/future compact and $U\subset M$ be open and
relatively compact. Let $O$ be some manifold and $F$ some vector bundle over
$O$. Then for each $n\in{\mathbb{N}}$, there is $m\in{\mathbb{N}}$ such that
for any $f\in\Gamma^{m}_{A\times O}(E\boxtimes F)$, we have
$G^{\pm}f|_{U\times O}\in\Gamma^{n}((E\boxtimes F)|_{U\times O})$, where
$G^{\pm}f(y,x):=G^{\pm}(f(\cdot,x))(y)$.

###### Proof.

As differentiability is local, it suffices to show this only on a chart domain
in $O$ on which $F$ is trivial. As a section is smooth if and only if its
composition with any bundle coordinate is smooth and the application of an
operator in the first component commutes with taking bundle coordinates in the
second component, it suffices to consider the case $O={\mathbb{R}}^{j}$,
$F={\mathbb{R}}^{j}\times{\mathbb{R}}$. Likewise, we may also assume that $U$
is contained in a chart with local trivialization for $E$ and thus partial
derivatives are well-defined.

For $n\in{\mathbb{N}}$, let $m\geq n$ be large enough such that for any $k\leq
n$, $G^{\pm}$ maps $\Gamma^{m-k}_{A}(E)$ to $\Gamma^{n-k}(E|_{U})$
continuously. This is possible by applying the previous proposition $n$ times.
We will inductively show the following in the domain of this chart:

######  Claim.

For any multiindices $\alpha$ and $\beta$ with $|\alpha|+|\beta|\leq k\leq n$,
we have

$\partial_{x}^{\beta}\partial_{y}^{\alpha}G^{\pm}(f(\cdot,x))(y)=\partial_{y}^{\alpha}G^{\pm}(\partial_{x}^{\beta}f(\cdot,x))(y)$

with the derivatives on the left hand side existing.

We proceed by induction over $k$, the case $k=0$ being immediate. Suppose we
have shown the statement for some $k<n$ and assume $|\alpha|+|\beta|\leq k$.
We have to show for every index $i$:

$\partial_{x_{i}}\partial_{y}^{\alpha}G^{\pm}(\partial_{x}^{\beta}f(\cdot,x))(y)=\partial_{y}^{\alpha}G^{\pm}(\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x))(y)$

(including existence of the derivative). We have for $h>0$ ($e_{i}$ denoting
the $i$-th standard unit vector) and any compact set $K$:

$\displaystyle\|\frac{\partial_{x}^{\beta}f(\cdot,x+he_{i})-\partial_{x}^{\beta}f(\cdot,x)}{h}-\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x)\|_{C^{m-|\beta|-1}(K)}$
$\displaystyle=\|\frac{1}{h}\int\limits_{0}^{h}\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x+te_{i})-\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x)dt\|_{C^{m-|\beta|-1}(K)}$
$\displaystyle\leq\sup\limits_{t<h}\|\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x+te_{i})-\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x)\|_{C^{m-|\beta|-1}(K)}.$

All derivatives of $\partial_{x_{i}}\partial_{x}^{\beta}f(y,x)$ of order up to
$m-|\beta|-1$ are continuous and hence uniformly continuous on the compact set

$K\times\\{x+te_{i}\mid t\in[0,1]\\}.$

Thus the end result of the above equation converges to $0$ for $h\rightarrow
0$. This means that the differential quotient

$\frac{\partial_{x}^{\beta}f(\cdot,x+he_{i})-\partial_{x}^{\beta}f(\cdot,x)}{h}$

converges to $\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x)$ in
$\Gamma^{m-|\beta|-1}_{A}(E)$. As $G^{\pm}$ maps $C^{m-|\beta|-1}$-sections to
$C^{n-|\beta|-1}$ sections continuously and $\partial_{y}^{\alpha}$ maps those
to $C^{0}$-sections (over $U\times O$) continuously, we have

$\displaystyle\partial_{x_{i}}\partial_{y}^{\alpha}G^{\pm}(\partial_{x}^{\beta}f(\cdot,x))(y)$
$\displaystyle=\lim\limits_{h\rightarrow
0}\partial_{y}^{\alpha}G^{\pm}\left(\frac{\partial_{x}^{\beta}f(\cdot,x+he_{i})-\partial_{x}^{\beta}f(\cdot,x)}{h}\right)(y)$
$\displaystyle=\partial_{y}^{\alpha}G^{\pm}\left(\lim\limits_{h\rightarrow
0}\frac{\partial_{x}^{\beta}f(\cdot,x+he_{i})-\partial_{x}^{\beta}f(\cdot,x)}{h}\right)(y)$
$\displaystyle=\partial_{y}^{\alpha}G^{\pm}(\partial_{x_{i}}\partial_{x}^{\beta}f(\cdot,x))(y),$

which finishes the proof of the claim.

To conclude the proof of the proposition, note that an arbitrary partial
derivative has the form

$\prod\limits_{j=0}^{l}\partial_{x}^{\beta_{j}}\partial_{y}^{\alpha_{j}}$

and iterated application of the claim yields that

$\prod\limits_{j=0}^{l}\partial_{x}^{\beta_{j}}\partial_{y}^{\alpha_{j}}G^{\pm}(f(\cdot,x))(y)=\prod\limits_{j=0}^{l}\partial_{y}^{\alpha_{j}}G^{\pm}(\prod\limits_{j=0}^{l}\partial_{x}^{\beta_{j}}f(\cdot,x))(y)$

exists and is continuous. ∎

We now introduce the precise form of the objects for which we want an
asymptotic expansion. Morally, they can be thought of as a rigorous version of

$\mathcal{K}(G^{\pm m})(\cdot,x).$

######  Definition 3.1.3.

For $x\in M$, $m\in\mathbb{Z}$ define distributions in
$\mathcal{D}^{\prime}(E\otimes E_{x}^{*})$ via

$G^{\pm m}_{x}:=(G^{\pm})^{m}\delta_{x}$

(where $\delta_{x}$ is the Dirac distribution at $x$, defined by
$\delta_{x}[\psi\otimes v]:=\psi(x)(v)$ for $\psi\in\Gamma_{c}(E)$ and $v\in
E^{*}_{x}$). In case $m=1$, we omit it:

$G^{\pm}_{x}:=G^{\pm 1}_{x}.$

######  Remark 3.1.4.

For negative $m$, we obtain Schwartz kernels for powers of $P$ (”evaluated” at
$x$) as both $G^{+}$ and $G^{-}$ are an inverse for $P$ on a suitable space.

Note that here we used the notation introduced at the end of 2.1.1, i.e.
$G^{\pm}$ acts as the identity in the $E^{*}_{x}$-part. There are multiple
other ways to think of $G^{\pm m}_{x}$. As we will show later, it is the
pullback of $\mathcal{K}(G)$ with respect to the inclusion of $M$ at $x$ (see
Lemma 6.1.5). Another way to see it is evaluation at $x$ composed with the
adjoint of $(G^{\pm})^{m}$:

######  Proposition 3.1.5.

For any $\psi\in\Gamma_{c}(E^{*})$, we have

$G^{\pm m}_{x}[\psi]=(G^{\mp}_{P^{*}})^{m}\psi(x)$

###### Proof.

Let $f_{n}$ be a sequence of test functions converging distributionally to
$\delta_{x}$. Then we have by Proposition 2.4.14 and continuity of $G^{\pm}$:

$\displaystyle G^{\pm m}_{x}(\psi)$
$\displaystyle=(G^{\pm})^{m}\delta_{x}(\psi)$
$\displaystyle=\lim_{n\rightarrow\infty}\int\limits_{M}\psi((G^{\pm})^{m}f_{n})$
$\displaystyle=\lim_{n\rightarrow\infty}\int\limits_{M}(G^{\mp}_{P^{*}})^{m}\psi(f_{n})$
$\displaystyle=\delta_{x}((G^{\mp}_{P^{*}})^{m}\psi)$
$\displaystyle=(G^{\mp}_{P^{*}})^{m}\psi(x).\qed$

To obtain the asymptotic expansion we want, we first need to restrict to a
suitable subset of $M$. Such subsets will be reffered to as GE sets (it is
left to the reader to decide whether this stands for ”Green’s expansion” or
for ”good enough”).

######  Definition 3.1.6.

A subset $U$ of $M$ shall be called GE, if it is open, geodesically convex,
globally hyperbolic, and causally compatible.

######  Remark 3.1.7.

Openness is required to restrict distributions to $U$. Geodesic convexity
ensures the existence of Riesz distributions and of Hadamard coefficients.
Global hyperbolicity ensures that $U$ has unique Green’s operators, while
causal compatibility ensures that those agree with the restriction of the
Green’s operators on $M$.

We shall make use of the following fact:

######  Proposition 3.1.8.

Every $p\in M$ has a basis of GE neighborhoods.

###### Proof.

This is basically the statement of [Min14, Corollary 2 and Remark 14], which
asserts the existence of a neighborhood basis of geodesically convex, globally
hyperbolic, causally convex sets. Causally convex means that for any
neighborhood $U$ from this basis, its intersection with every causal curve is
connected. We only need to show that this implies causal compatibility, which
is not too difficult: If $y\in J_{\pm}^{M}(x)$ for $x\in U$, there must be a
future/past directed causal curve $\gamma$ in $M$ that connects $x$ and $y$.
As the intersection of $\gamma$ with $U$ is connected and contains $x$ and
$y$, it must also contain the segment between $x$ and $y$, so the two can be
joined by a future/past directed causal curve in $U$. Thus $x\in
J_{\pm}^{U}(y)$ and hence $U$ is causally compatible. ∎

We can now prove the main theorem of this section

######  Theorem 3.1.9.

Let $U\subseteq M$ be a GE set and let $m\in\mathbb{Z}$. Then we have the
asymptotic expansion

$G^{\pm
m}_{x}|_{U}\sim_{x}\sum\limits_{k=0}^{\infty}\binom{m+k-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x).$

For $m<0$, the right hand side is a finite sum, as all summands with $k+m>0$
vanish, and we have equality.

######  Remark 3.1.10.

Note that the special case $m=1$ reproduces the expansion of [BGP07,
Proposition 2.5.1], though on slightly different domains (this means that even
though they are morally the same, it is not easy to reduce either one to the
other). Unlike in [BGP07], there is no assumption on the volume of $U$ and it
may be chosen independently of $P$.

######  Remark 3.1.11.

In case $m<0$, this is actually a formula for the kernel of powers of $P$. All
even Riesz distributions of order $0$ or less are the same in the advanced and
retarded case and supported only at the diagonal, so this expansion is not as
surprising as it might seem on first glance. From the case $m=-1$, we obtain
that

$\mathcal{K}(P)(\cdot,x)=V^{0}_{x}R(-2,x)-V^{1}_{x}\delta_{x}.$

In particular, $P$ is uniquely determined by the metric $g$ (which determines
the Riesz-distributions) and the first two Hadamard coefficients $V^{0}$ and
$V^{1}$.

###### Proof.

We proceed by two-way induction on $m$, showing that if the statement holds
for $m\in\mathbb{Z}$, it also holds for $m+1$ if $m\geq 0$ and for $m-1$, if
$m\leq 0$. The case $m=0$ follows from $R_{\pm}(0,x)=\delta_{x}$. Fix $x\in
U$.

We make some preliminary calculations. For $k+m\neq 0$ we can use the
characterization of $\rho_{x}^{U}$ (2.5.9) and the transport equations (see
2.5.10) to calculate

$\displaystyle\binom{k+m}{k}P(V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x))$
$\displaystyle=\binom{k+m}{k}\Big{(}\frac{-1}{2k+2m}(\rho_{x}^{U}-2k-2m)V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)+(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x)\Big{)}$
$\displaystyle=\binom{k+m}{k}\Big{(}\frac{-1}{2k+2m}((2kPV^{k-1,U}_{x}-2mV^{k,U}_{x})R^{U}_{\pm}(2k+2m,x))$
$\displaystyle+(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x)\Big{)}$
$\displaystyle=\binom{k+m}{k}\frac{2m}{2k+2m}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)-\binom{k+m}{k}\frac{2k}{2m+2k}(PV^{k-1,U}_{x})R^{U}_{\pm}(2k+2m,x)$
$\displaystyle+\binom{k+m}{k}(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x)$
$\displaystyle=\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)-\binom{k+m-1}{k-1}(PV^{k-1,U}_{x})R^{U}_{\pm}(2k+2m,x)$
$\displaystyle+\binom{k+m}{k}(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x))$

In the special case $k=m=0$, we obtain:

$\displaystyle P(V^{0,U}_{x}R_{\pm}(2,x))$
$\displaystyle=\lim\limits_{\alpha\rightarrow
0}P(V^{0,U}_{x}R_{\pm}(\alpha+2,x))$
$\displaystyle=\lim\limits_{\alpha\rightarrow
0}P(V^{0,U}_{x})R_{\pm}(\alpha+2,x)-(\frac{1}{\alpha}\rho_{x}^{U}V^{0,U}_{x}-V^{0,U}_{x})R_{\pm}(\alpha,x)$
$\displaystyle=\lim\limits_{\alpha\rightarrow
0}P(V^{0,U}_{x})R_{\pm}(\alpha+2,x)+V^{0,U}_{x}R_{\pm}(\alpha,x)$
$\displaystyle=P(V^{0,U}_{x})R_{\pm}(2,x)+V^{0,U}_{x}R_{\pm}(0,x)$
$\displaystyle=P(V^{0,U}_{x})R_{\pm}(2,x)+V^{0,U}_{x}(x)\delta_{x}$
$\displaystyle=\delta_{x}+P(V^{0,U}_{x})R_{\pm}(2,x).$

either way, we obtain for $k=0$:

$P(V^{0,U}_{x}R^{U}_{\pm}(2m+2,x))=V^{0,U}_{x}R^{U}_{\pm}(2m,x)+(PV^{0,U}_{x})R^{U}_{\pm}(2m+2,x)$

Putting everything together, for any $N\in{\mathbb{N}}$ in case $m\geq 0$ and
for any $N<-m$ if $m<0$, we have

$\displaystyle
P\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)$
$\displaystyle=V^{0,U}_{x}R^{U}_{\pm}(2m,x)+(PV^{0,U}_{x})R^{U}_{\pm}(2m+2,x)+\sum\limits_{k=1}^{N}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)$
$\displaystyle-\binom{k+m-1}{k-1}(PV^{k-1,U}_{x})R^{U}_{\pm}(2k+2m,x)+\binom{k+m}{k}(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)-\sum\limits_{k=0}^{N-1}\binom{k+m}{k}(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x)$
$\displaystyle+\sum\limits_{k=0}^{N}\binom{k+m}{k}(PV^{k,U}_{x})R^{U}_{\pm}(2k+2m+2,x))$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)+\binom{N+m}{N}(PV^{N,U}_{x})R^{U}_{\pm}(2N+2m+2,x)$
$\displaystyle=:\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)+E_{N}(\cdot,x)$

where

$E_{N}(\cdot,x):=\binom{N+m}{N}(PV^{N,U}_{x})R^{U}_{\pm}(2N+2m+2,x)$

is $C^{n}$ in both variables for $N\geq\frac{d}{2}+n-m$.

Case 1: $m+1\Rightarrow m$, $m<0$
Suppose for induction that

$G^{\pm
m+1}_{x}=\sum\limits_{k=0}^{-m-1}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)$

(This implies the theorem for $m+1$, as all further summands are 0). Let
$N=-m-1$. Using the fact (see [BGP07, below Proposition 2.3.1] that

$PV^{N,U}_{x}(x)=-V^{N+1,U}_{x}(x)$

and the identity

$\binom{-a}{b}=(-1)^{b}\binom{a+b-1}{b}$

for integer $b$, we calculate

$\displaystyle E_{N}(\cdot,x)$
$\displaystyle=\binom{-1}{N}(PV^{N,U}_{x})R^{U}_{\pm}(0,x)$
$\displaystyle=(-1)^{N}\binom{N}{N}(PV^{N,U}_{x})\delta_{x}$
$\displaystyle=(-1)^{N}PV^{N,U}_{x}(x)\delta_{x}$
$\displaystyle=(-1)^{N+1}V^{N+1,U}_{x}(x)\delta_{x}$
$\displaystyle=(-1)^{N+1}\binom{N+1}{N+1}V^{N+1,U}_{x}(x)\delta_{x}$
$\displaystyle=\binom{-1}{N+1}V^{N+1,U}_{x}R^{U}_{\pm}(0,x)$
$\displaystyle=\binom{-1}{-m}V_{x}^{-m,U}R_{\pm}^{U}(0,x).$

We obtain from the inductive hypothesis and our previous calculations:

$\displaystyle G^{\pm m}_{x}$ $\displaystyle=PG^{\pm m+1}_{x}$
$\displaystyle=P\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)+E_{N}(\cdot,x)$
$\displaystyle=\sum\limits_{k=0}^{-m-1}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)+\binom{-1}{-m}V^{-m,U}_{x}R^{U}_{\pm}(0,x)$
$\displaystyle=\sum\limits_{k=0}^{-m}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)$

Case 2: $m\Rightarrow m+1$, $m\geq 0$
Fix $n\in{\mathbb{N}}$ and assume the theorem holds for some $m\geq 0$. We
first show that the expansion holds on relatively compact GE subsets of $U$.
Let $W,O\subseteq U$ be relatively compact (in $U$) and GE. Proposition 3.1.2
implies that there is $n^{\prime}\in{\mathbb{N}}$ such that $G^{\pm}_{P|_{U}}$
maps $\Gamma^{n^{\prime}}_{J^{U}_{\pm}(\overline{O})\times O}(E|_{U}\boxtimes
E^{*}|_{U})$ to sections that are $C^{n}$ on $W\times O$. Assume $x\in O$. By
the inductive hypothesis, we may choose $N\geq\frac{d}{2}+n^{\prime}$ such
that

$F_{N}(\cdot,x):=G^{\pm
m}_{x}|_{U}-\sum\limits_{k=0}^{N}\binom{m+k-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)$

defines a $C^{n^{\prime}}$-section $F_{N}$ on $U$. We then have

$\displaystyle G^{\pm m+1}_{x}|_{U}$ $\displaystyle=(G^{\pm}_{P}G^{\pm
m}_{x})|_{U}$ $\displaystyle=G^{\pm}_{P|_{U}}G^{\pm m}_{x}|_{U}$
$\displaystyle=G^{\pm}_{P|_{U}}\left(\sum\limits_{k=0}^{N}\binom{m+k-1}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m,x)+F_{N}(\cdot,x)\right)$
$\displaystyle=G^{\pm}_{P|_{U}}\left(P\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)-E_{N}(\cdot,x)+F_{N}(\cdot,x)\right)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)+G^{\pm}_{P|_{U}}(F_{N}(\cdot,x)-E_{N}(\cdot,x))$
$\displaystyle=:\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)+\tilde{F}_{N}(\cdot,x).$

with

$\tilde{F}_{N}(\cdot,x):=G^{\pm
m+1}_{x}|_{U}-\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)=G^{\pm}_{P|_{U}}(F_{N}(\cdot,x)-E_{N}(\cdot,x)).$

By our choice of ${n^{\prime}}$ and $N$, $\tilde{F}_{N}$ is $C^{n}$ on
$W\times O$, as both $E_{N}|_{U\times O}$ and $F_{N}|_{U\times O}$ are
$C^{n^{\prime}}$ and supported in $J_{\pm}(O)\times O$. Note that the only
choices we made after choosing $W$ and $O$ (apart from assuming $x\in O$) were
that of ${n^{\prime}}$ and $N$. We thus want to make the expansion independent
of the choice of $N$.

Let $N_{0}\geq\frac{d}{2}+n$ be arbitrary. For $k\geq N_{0}$, all Riesz
distributions $R^{U}_{\pm}(2k+2m+2,x)$ are given by $C^{n}$-functions. We see
that

$\displaystyle\tilde{F}_{N_{0}}(\cdot,x)$ $\displaystyle=G^{\pm
m+1}_{x}|_{U}-\sum\limits_{k=0}^{N_{0}}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)$
$\displaystyle=\sum\limits_{k=N_{0}+1}^{N}\binom{m+k}{k}V^{k,U}_{x}R^{U}_{\pm}(2k+2m+2,x)+\tilde{F}_{N}(\cdot,x)$

is $C^{n}$ on $W\times O$. As this is independent of $W$ and $O$, these can
replaced by arbitrary relatively compact GE sets. All points in $U$ have
relatively compact GE neighborhoods (choose any relatively compact
neighborhood and then choose a GE neighborhood contained in it by Proposition
3.1.8). Thus any $(y,x)\in U\times U$ has a neighborhood on which
$\tilde{F}_{N_{0}}$ is $C^{n}$, which means it is $C^{n}$ on all of $U\times
U$. As $n$ was arbitrary, we have the desired asymptotic expansion for $m+1$
and have thus concluded our induction. ∎

###  3.2 Green’s “resolvent”

We now consider the Green’s operators for the operator $P-z$ for
$z\in{\mathbb{C}}$, which is still normally hyperbolic. If we view
$G^{\pm}_{P}$ as something like an inverse for $P$, then $G^{\pm}_{P-z}$ is
something like a resolvent. We seek to derive an expansion similar to the one
above, where only the Hadamard coefficients of $P$ rather than $P-z$ appear
and the $z$-dependence is instead shifted to a generalzed version of Riesz
distributions. For this section, fix a convex open set $U$ and let $p\in U$ be
some point. Any Hadamard coefficients, Riesz distributions and similar objects
depending on a convex open subset will be defined on $U$ unless otherwise
indicated and we will omit the superscript $U$.

######  Definition 3.2.1.

Let $\square_{V}$ denote the d’Alembertian on a Lorentzian vector space $V$.

For $z\in{\mathbb{C}}$ and $m\in{\mathbb{N}}$, we define resolvent Riesz
distributions as

$R_{\pm}(z,2m,x):=(\exp_{x})^{-1*}(G^{\pm m}_{\square_{V}-z,0})$

The motivation for this is that it reproduces the original Riesz-distributions
in the case $z=0$:

######  Proposition 3.2.2.

On a Lorentzian vector space $V$, we have

$R^{V}_{\pm}(2m)=G^{\pm m}_{\square_{V},0}.$

Thus we have

$R_{\pm}(0,2m,x)=R_{\pm}(2m,x).$

###### Proof.

We have ([BGP07, Lemma 1.2.2])

$\square_{V}R_{\pm}^{V}(2m+2)=R_{\pm}^{V}(2m)$

and hence

$R_{\pm}^{V}(2m+2)=G^{\pm}_{\square_{V}}\square_{V}R_{\pm}^{V}(2m+2)=G^{\pm}_{\square_{V}}R_{\pm}^{V}(2m).$

Together with

$R_{\pm}^{V}(0)=\delta_{0},$

we find that the $R_{\pm}^{V}(2m)$ satisfy the recursion defining $G^{\pm
m}_{\square_{V},0}$, hence we have shown the first part.

The second part follows by inserting the first part into the definition of the
Riesz distributions on $U$ and comparing with that of the resolvent Riesz
distributions. ∎

We want to exploit the asymptotic expansion that we already have in order to
obtain the one we would like to get. For this, we need to know the Hadamard
coefficients for $P-z$ in terms of those for $P$.

######  Definition 3.2.3.

For $z\in{\mathbb{C}}$, let $V^{k}_{x}(z)$ be the corresponding Hadamard
coefficients for $P-z$.

######  Proposition 3.2.4.

We have

$V^{k}_{x}(z)=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}V^{k-m}_{x}.$

######  Remark 3.2.5.

The corresponding formula for the heat coefficients

$a_{k}(\Delta-z)=\sum\limits_{m=0}^{k}\frac{1}{m}z^{m}a_{k-m}(\Delta)$

can be shown analogously. Alternatively, that formula can also be deduced by
Taylor expanding $e^{tz}$ and multiplying out the expansions in

$e^{-t(\Delta-z)}=e^{tz}e^{-t\Delta}.$

###### Proof.

Let

$V_{k}(z):=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}V^{k-m}_{x}$

We need to show that $V_{0}(z)(x,x)=1$ and the $V_{k}(z)$ satisfy the
transport equations. The former holds, as $V_{0}(z)=V^{0}_{x}$. For the
latter, we calculate using the transport equation for $P$:

$\displaystyle(\rho_{x}^{U}-2k)V_{k}(z)$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}(\rho_{x}^{U}-2k)V^{k-m}_{x}$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}((\rho_{x}^{U}-2(k-m))V^{k-m}_{x}-2mV^{k-m}_{x})$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}2(k-m)PV^{k-m-1}_{x}-\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}2mV^{k-m}_{x}$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k-1}{m}z^{m}2kPV^{k-m-1}_{x}-\sum\limits_{m=1}^{k}\binom{k-1}{m-1}z^{m}2kV^{k-m}_{x}$
$\displaystyle=\sum\limits_{m=0}^{k-1}\binom{k-1}{m}z^{m}2kPV^{k-m-1}_{x}-\sum\limits_{m=0}^{k-1}\binom{k-1}{m}z^{m+1}2kV^{k-m-1}_{x}$
$\displaystyle=2k(P-z)\sum\limits_{m=0}^{k-1}\binom{k-1}{m}z^{m}V^{k-m-1}_{x}$
$\displaystyle=2k(P-z)V_{k-1}(z).$

The $V_{k}(z)$ thus satisfy the transport equations for $P-z$, so they are the
desired Hadamard coefficients. ∎

The other ingredient we need is an expansion for the resolvent Riesz
distributions in terms of the standard ones.

######  Lemma 3.2.6.

On Minkowski space, we have the asymptotic expansion

$G^{\pm
k}_{\square-z,0}\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}z^{m}R_{\pm}(2m+2k).$

###### Proof.

As $R_{\pm}(2m+2k)=R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k,0)$, Theorem 3.1.9 applied
to $\square-z$ gives us the asymptotic expansion

$G^{\pm
k}_{\square-z,0}\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}V_{m}(z)R_{\pm}(2m+2k),$

where the $V_{m}(z)$ are the Hadamard coefficients at $0$ for $\square-z$,
i.e. the unique solutions to

$V_{0}(z)(0)=1$

and

$\rho_{0}^{{{\mathbb{R}}^{d}}}V_{m}(z)-2mV_{m}(z)=2m(\square-z)V_{m-1}(z).$

These equations are solved by the constant functions $V_{m}(z):=z^{m}$, as

$\rho_{0}^{{\mathbb{R}}^{d}}z^{m}=\left<\operatorname{grad}\gamma,\operatorname{grad}z^{m}\right>+(\square\gamma-d)z^{m}=0$

and

$\square z^{m}=0,$

so we obtain

$G^{\pm
k}_{\square-z,0}\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}z^{m}R_{\pm}(2m+2k).\qed$

This carries over from Minkowski space to $M$ in a straightforward way:

######  Lemma 3.2.7.

We have the asymptotic expansion

$R_{\pm}(z,2k,x)\sim_{x}\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}z^{m}R^{U}_{\pm}(2m+2k,x).$

###### Proof.

As $U$ is convex, it is contractible and thus $TU$ is trivial. Let $\phi\colon
U\times{\mathbb{R}}^{d}\rightarrow TU$ be an isometric trivialization that
preserves time-orientation. As $\phi$ is a time-orientation preserving
isometry on every fibre, it preserves all objects defined only in terms of the
metric and time-orientation, i.e.

$\phi_{x}^{-1*}(R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k))=R^{T_{x}M}_{\pm}(2m+2k)$

and

$\phi_{x}^{-1*}(G^{\pm k}_{\square_{{\mathbb{R}}^{d}}-z,0})=G^{\pm
k}_{\square_{T_{x}M}-z,0}$

Let

$\operatorname{Exp}\colon\operatorname{Dom}(\operatorname{Exp})\subseteq
TU\rightarrow U\times U$

denote the exponential map with variable basepoint, i.e.
$\operatorname{Exp}(v)=(x,\exp_{x}(v))$ for $v\in T_{x}U$. This is smooth.

Let $n\in{\mathbb{N}}$ be arbitrary. For $N$ large enough such that

$G^{\pm
k}_{\square_{{\mathbb{R}}^{d}}-z,0}=\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k)+F$

with $F\in C^{n}({\mathbb{R}}^{d})$, we have

$\displaystyle R_{\pm}(z,2k,x)$ $\displaystyle=(\exp_{x})^{-1*}(G^{\pm
k}_{\square_{T_{x}M}-z,0})$
$\displaystyle=(\exp_{x}\circ\phi_{x})^{-1*}(G^{\pm
k}_{\square_{{\mathbb{R}}^{d}}-z,0})$
$\displaystyle=(\exp_{x}\circ\phi_{x})^{-1*}\left(\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k)+F\right)$
$\displaystyle=\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}(exp_{x})^{-1*}R^{T_{x}M}_{\pm}(2m+2k)+F\circ\phi_{x}^{-1}\circ
exp_{x}^{-1}$
$\displaystyle=\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}R^{U}_{\pm}(2m+2k,x)+F\circ\phi^{-1}\circ\operatorname{Exp}^{-1}(\cdot,x).$

as $F\circ\phi^{-1}\circ\operatorname{Exp}^{-1}$ is $C^{n}$ and $n$ was
arbitrary, we have the desired expansion. ∎

We now have everything in place to prove the main result of this section: If,
in the asymptotic expansions for the Green’s operators of $P$, we replace the
standard Riesz distributions with the resolvent Riesz distributions for
$z\in{\mathbb{C}}$, then we obtain asymptotic expansions for the Green’s
operators of $P-z$.

######  Theorem 3.2.8.

If $U\subseteq M$ is GE, we have the following asymptotic expansion in $U$:

$G^{\pm}_{P-z,x}|_{U}\sim_{x}\sum\limits_{k=0}^{\infty}V^{k,U}_{x}R^{U}_{\pm}(z,2k+2,x).$

More generally,

$G^{\pm
m}_{P-z,x}|_{U}\sim_{x}\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}V^{k,U}_{x}R^{U}_{\pm}(z,2k+2m,x).$

###### Proof.

Applying Theorem 3.1.9 to $P-z$ and inserting our formula for the
$z$-dependent Hadamard coefficients (Proposition 3.2.4), we obtain

$\displaystyle G^{\pm m}_{P-z,x}|_{U}$
$\displaystyle\sim_{x}\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}V^{k}_{x}(z)R_{\pm}(2k+2m,x)$
$\displaystyle=\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}\sum\limits_{l+n=k}\binom{k}{n}z^{n}V^{l}_{x}R_{\pm}(2k+2m,x)$
$\displaystyle=\sum\limits_{l=0}^{\infty}\sum\limits_{n=0}^{\infty}\binom{n+l+m-1}{n+l}\binom{n+l}{n}z^{n}V^{l}_{x}R_{\pm}(2(n+m+l),x)$
$\displaystyle=\sum\limits_{l=0}^{\infty}\sum\limits_{n=0}^{\infty}\binom{n+l+m-1}{n}\binom{l+m-1}{l}z^{n}V^{l}_{x}R_{\pm}(2(n+m+l),x)$
$\displaystyle=\sum\limits_{l=0}^{\infty}\binom{l+m-1}{l}V^{l}_{x}\sum\limits_{n=0}^{\infty}\binom{n+l+m-1}{n}z^{n}R_{\pm}(2(n+m+l),x)$
$\displaystyle\sim_{x}\sum\limits_{l=0}^{\infty}\binom{l+m-1}{l}V^{l}_{x}R_{\pm}(z,2(m+l),x).$

Here we used the identity

$\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$

and Lemma 3.2.7.

This proves the second claim, the first claim follows as the special case
$m=1$. ∎

##  Chapter 4 Constructing an action-like function

In this chapter we will take our first step towards extracting the Hadamard
coefficients from the asymptotic expansion we have seen. We will construct a
function from the Green’s operator that has an asymptotic expansion in terms
of the Hadamard coefficients. This is similar to an analogue of a spectral
action, but with additional terms coming from the Hadamard coefficients’
derivatives.

For technical reasons, we’ll be looking not at the two Green’s operators
individually, but at their difference, sometimes referred to as the causal
propagator.

######  Definition 4.0.1.

Set

$G:=G^{+}-G^{-},$ $G_{x}:=G^{+}_{x}-G^{-}_{x},$
$R(\alpha):=R_{+}(\alpha)-R_{-}(\alpha)$

and

$R^{U}(\alpha,x):=R^{U}_{+}(\alpha,x)-R^{U}_{-}(\alpha,x).$

Taking the difference of the asymptotic expansions for $G^{\pm}_{x}$, we
obtain

$G_{x}|_{U}\sim_{x}\sum\limits_{k=0}^{\infty}V^{k,U}_{x}R^{U}(2k+2,x).$

We are motivated by the following formal calculation:

Let $w$ be a timelike geodesic in $M$ and let $x=w(0)$. Then we have (for
$\Re(\alpha)$ large enough)

$R^{U}(\alpha,x)(w(t))=c_{\alpha}|t|^{\alpha-d}\operatorname{sign}(t)$

Writing

$\tilde{V}_{k}(t):=V^{k,U}_{x}(w(t)),$

we can taylor expand

$\tilde{V}_{k}(t)\sim\sum\limits_{n=0}^{\infty}\frac{1}{n!}\tilde{V}_{k}^{(n)}(0)t^{n}.$

For some function $f\in C_{c}^{\infty}({\mathbb{R}})$ and $s>0$, we can then
calculate (formally):

$\displaystyle\int\limits_{\mathbb{R}}G_{x}(w(t))f(\tfrac{t}{s})dt$
$\displaystyle\sim\sum\limits_{k=0}^{\infty}\int\limits_{\mathbb{R}}V^{k,U}_{x}(w(t))R^{U}(2k+2,x)(w(t))f(\tfrac{t}{s})dt$
$\displaystyle=\sum\limits_{k=0}^{\infty}\int\limits_{\mathbb{R}}\tilde{V}_{k}(t)c_{2k+2}t^{2k+2-d}\operatorname{sign}(t)f(\tfrac{t}{s})dt$
$\displaystyle\sim\sum\limits_{k=0}^{\infty}c_{2k+2}\int\limits_{\mathbb{R}}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\tilde{V}_{k}^{(n)}(0)t^{2k+n+2-d}\operatorname{sign}(t)f(\tfrac{t}{s})dt$
$\displaystyle=\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{\infty}\frac{c_{2k+2}}{n!}\tilde{V}_{k}^{(n)}(0)s^{2k+n+3-d}\int\limits_{\mathbb{R}}\frac{1}{s}\left(\frac{t}{s}\right)^{2k+n+2-d}\operatorname{sign}(\tfrac{t}{s})f(\tfrac{t}{s})dt$
$\displaystyle=\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{\infty}\frac{c_{2k+2}}{n!}\tilde{V}_{k}^{(n)}(0)s^{2k+n+3-d}\int\limits_{\mathbb{R}}t^{2k+n+2-d}\operatorname{sign}(t)f({t})dt$
$\displaystyle=:\sum\limits_{k=0}^{\infty}\sum\limits_{n=0}^{\infty}a_{k,n}(f)\tilde{V}_{k}^{(n)}(0)s^{2k+n+3-d}$

with some coefficients $a_{k,n}(f)$ independent of $s$, $M$ and $P$ (except
through the dimension). From this, or something similar, one might hope to
extract the values of the Hadamard coefficients at the diagonal, i.e.
$\tilde{V}_{k}(0)$. The problem of performing this extraction is postponed to
the next chapter, as this chapter is devoted to making the above calculation
precise. The main problem in this is that, for small $k$, the Riesz-
distributions are not given by functions, so evaluation at $w(t)$ is ill-
defined.

###  4.1 Holomorphicity of Riesz distributions for some wavefront

While evaluating the Riesz-distributions at a single point is problematic, we
will show that restricting them to a timelike geodesic is possible. By
wavefront calculus, it suffices to show that the Riesz-distributions’
wavefront does not intersect the conormal bundle of the curve. To actually
compute what the restriction does, we also need that the Riesz-distributions
are holomorphic in the space of distributions with a suitable wavefront, so
that the restriction is still holomorphic. Showing these two things is the
goal of this section.

The conormal bundle of a timelike geodesic consists of spacelike vectors.
Therefore we want to show that the wavefront set of the Riesz distributions
does not contain spacelike vectors. This is the point where we have to use
that we work with the difference $R=R_{+}-R_{-}$, rather than the advanced or
retarded Riesz distributions individually. In this difference, the worst
singularities at $0$ will cancel out. As wavefront calculus is invariant under
diffeomorphisms, it will be sufficient to work on Minkowski space.

We thus consider a spacelike vector $\xi\in{\mathbb{R}}^{d}$. Deciding whether
$\xi$ is in the wavefront set of a function involves integration against
$e^{-i\left<\xi,\cdot\right>}$. Our strategy is to find, for each
$x\in{\mathbb{R}}^{d}$, another point $O_{\xi}(x)$ such that