Wick rotation in the lapse: admissible complex metrics

and the associated heat kernel

Rudrajit Banerjee111email<EMAIL_ADDRESS>and Max
Niedermaier222email<EMAIL_ADDRESS>
1Okinawa Institute of Science and Technology Graduate University,
1919-1, Tancha, Onna, Kunigami District
Okinawa 904-0495, Japan

2Department of Physics and Astronomy
University of Pittsburgh, 100 Allen Hall
Pittsburgh, PA 15260, USA

Abstract

> A Wick rotation in the lapse (not in time) is introduced that interpolates
> between Riemannian and Lorentzian metrics on real manifolds admitting a co-
> dimension one foliation. The definition refers to a fiducial foliation but
> covariance under foliation changing diffeomorphisms is ensured. In
> particular, the resulting complex metrics are admissible in the sense of
> Kontsevich-Segal in all (fiducial and non-fiducial) foliations. This setting
> is used to construct a Wick rotated heat semigroup, which remains well-
> defined into the near Lorentzian regime. Among the results established for
> the Wick rotated version are: (i) Existence as an analytic semigroup
> uniquely determined by its sectorial generator. (ii) Construction of an
> integral kernel that is jointly smooth in the semigroup time and both
> spacetime arguments. (iii) Existence of an asymptotic expansion for the
> kernel’s diagonal in (shifted) powers of the semigroup time whose
> coefficients are the Seeley-deWitt coefficients evaluated on the complex
> metrics. (iv) Emergence of a Schrödinger evolution group in the strict
> Lorentzian limit. The toolbox includes local regularity results for
> admissible complex metrics.

###### Contents

  1. 1 Introduction
  2. 2 Wick rotated lapse and admissible complex metrics
    1. 2.1 Foliation changing diffeomorphisms
    2. 2.2 Admissibility criteria for scalar field theories.
  3. 3 Wick rotated heat semigroup from complexified Hessian
    1. 3.1 Background: analytic semigroups
    2. 3.2 Spectrum and sectoriality of the complexified Hessian
  4. 4 Semigroup action and the Wick rotated heat kernel
  5. 5 Asymptotic expansion of diagonal Wick rotated heat kernel
  6. 6 Schrödinger evolution group from the $\theta\rightarrow 0^{+}$ limit
  7. 7 Conclusions
  8. A Foliation geometry
  9. B Distributions and Sobolev spaces
  10. C Local regularity results

## 1\. Introduction

For field theories on curved non-stationary backgrounds the notion of a Wick
rotation is problematic. The proposed approaches include: rank one
deformations [6, 45], complex analytic metrics [34, 20, 48], and Vielbein
formulations [31, 43, 29, 46]. They have different range of applicability,
limitations, and occasionally overlap. For example, a Wick rotation in time
may be limited to purely electric metrics [25]; much of the Vielbein analysis
is so far pointwise without change of chart. A recent survey [1] deems none of
the existing proposals fully satisfactory.

Here we explore a notion of a Wick rotation on $1\\!+\\!d$ dimensional real
smooth manifolds $M$ that admit a codimension-one foliation
$t\mapsto\Sigma_{t}$ into $d$ dimensional leaves which are level surfaces
$T=t$ of a scalar function $T$. Importantly, the atlas of charts of the
manifold $M$ are kept real and merely some of the metric components are
complexified. The diffeomorphism group changing charts remains real throughout
(and we take it to consist of smooth maps connected to the identity that are
orientation- and boundary preserving). In adapted coordinates
$y^{\mu}=(t,x^{a})$ the Lorentzian and Euclidean metrics to be related may
then be parameterized according to

$ds^{2}_{\epsilon_{g}}=g^{\epsilon_{g}}_{\mu\nu}(y)dy^{\mu}dy^{\nu}=\epsilon_{g}N^{2}dt^{2}+{\rm
g}_{ab}(dx^{a}+N^{a}dt)(dx^{b}+N^{b}dt)\,,$ (1.1)

where $N$ is the lapse, $N^{a}$ the shift, and ${\rm g}_{ab}$ the metric on
$\Sigma_{t}$. We collect these fields into a triple $(N,N^{a},{\rm
g}_{ab})_{\epsilon_{g}}$, where the subscript indicates the signature of the
metric reconstructed from these data. The sign of the signature parameter
$\epsilon_{g}=\pm 1$ cannot be flipped along a real path in $[-1,1]$ without
encountering degenerate metrics. Instead, we use in a fiducial foliation a
Wick rotation in the lapse:

$(N,N^{a},{\rm
g}_{ab})_{\epsilon_{g}}\mapsto(i\epsilon_{g}^{-1/2}e^{-i\theta}N,N^{a},{\rm
g}_{ab})_{\epsilon_{g}}\,,\quad\theta\in[0,\pi)\,,$ (1.2)

where $\sqrt{\epsilon_{g}}=+1,i$ for $\epsilon_{g}=1,-1$. Crucially, the time
coordinate remains real; it is the lapse field $N(t,x)$ in the reference
foliation that is complexified. The conventions are such that starting from
either initial signature the line element after (1.2) is
$ds^{2}_{\theta}=-e^{-2i\theta}N^{2}dt^{2}+{\rm
g}_{ab}(dx^{a}+N^{a}dt)(dx^{b}+N^{b}dt)$. Thus Lorentzian and Euclidean
signature are recovered by the $\theta\rightarrow 0^{+}$ and
$\theta\rightarrow\pi/2$ limits, respectively, irrespective of the initial
signature.

In the first part of the article we argue that (1.2) gives rise to a fairly
satisfactory notion of a Wick rotation. In particular, it can consistently be
transferred to foliations other than the fiducial one. The resulting complex
metric $g^{\theta}$ satisfies the admissibility criterion for complex metrics
proposed in [31, 29], trivially in the reference foliation and nontrivially in
all others.

In the second part we show that in this framework a Wick rotated heat semi-
group exists that remains well-defined into the near Lorentzian regime. The
generator of the semigroup $\Delta_{\theta}$ can be written in terms of the
Euclidean/Lorentzian signature Hessian ${\cal D}_{\pm}=-\nabla_{\pm}^{2}+V$
(for a nonnegative smooth potential $V$) as

$\Delta_{\theta}=-\sin\theta\,{\cal D}_{+}-i\cos\theta\,{\cal
D}_{-},\quad\theta\in(0,\pi)\,,$ (1.3)

where in general $[{\cal D}_{+},{\cal D}_{-}]\neq 0$. For convenient
orientation we summarize our main results here in a slightly cursory form. The
precise definitions and conditions can be found in the corresponding theorems.

  * (a)

$\Delta_{\theta}$ is an unbounded operator on a Sobolev domain dense in
$L^{2}$. The same holds for its adjoint $\Delta_{\theta}^{*}$, and
$\Delta_{\theta}^{*}=\Delta_{\pi-\theta}$, including domains.

  * (b)

The spectrum of $\Delta_{\theta}$ is contained in a wedge of the left half
plane, $|{\rm Arg}\lambda|\geq\pi/2+\tilde{\theta}$, with
$\tilde{\theta}:=\min\\{\theta,\pi-\theta\\}$.

  * (c)

The resolvent $[z-\Delta_{\theta}]^{-1}\\!$, $|{\rm
Arg}(z)|<\pi/2\\!+\\!\tilde{\theta}$, obeys norm bounds that qualify
$\Delta_{\theta}$ as the generator of an analytic semigroup, $\zeta\mapsto
e^{\zeta\Delta_{\theta}}$, with $|{\rm Arg}\zeta|<\tilde{\theta}$.

  * (d)

$\zeta\mapsto e^{\zeta\Delta_{\theta}}$ is a bounded analytic semigroup on
$L^{2}$, which is uniquely determined by $\Delta_{\theta}$ (including domain)
and contractive, $\|e^{\zeta\Delta_{\theta}}\|\leq 1$, $|{\rm
Arg}(\zeta)|<\tilde{\theta}$ ($\zeta>0$ in particular).

  * (e)

The operator $e^{\zeta\Delta_{\theta}}$, $|{\rm Arg}(\zeta)|<\tilde{\theta}$,
acts as an integral operator on $L^{2}$ functions with a kernel
$K_{\zeta}^{\theta}(t,x;t^{\prime},x^{\prime})$ that is jointly smooth in
$(\zeta,t,x,t^{\prime},x^{\prime})$, and obeys
$K_{\zeta}^{\theta}(t,x;t^{\prime},x^{\prime})^{*}=K_{\zeta^{*}}^{\pi-\theta}(t^{\prime},x^{\prime};t,x)$.

  * (f)

The diagonal kernel admits an asymptotic expansion of the form

$\displaystyle\\!K_{\zeta}^{\theta}(t,x;\\!t,x)\asymp\frac{(-ie^{i\theta})^{\frac{d-1}{2}}}{(4\pi\zeta)^{\frac{d+1}{2}}}\sum_{n\geq
0}A_{n}^{\theta}(t,x)(ie^{-i\theta}\zeta)^{n},$

where $A_{n}^{\theta}$ are the heat kernel coefficients evaluated on
$g^{\theta}$.

  * (g)

The $\theta\rightarrow 0^{+}$ limit gives rise to well-defined Schrödinger
evolution group. Specifically, whenever $e^{-is\overline{{\cal D}}_{-}}$,
$s\in{\mathbb{R}}$, is generated by the unique self-adjoint closure of the
Lorentzian Hessian ${\mathcal{D}}_{-}$ defined on $C_{c}^{\infty}(M)$, one has
$e^{s\Delta_{\theta}}\xrightarrow{\theta\rightarrow
0^{+}}e^{-is\overline{{\cal D}}_{-}}$ and
$e^{s\Delta_{\pi-\theta}}\xrightarrow{\theta\rightarrow
0^{+}}e^{+is\overline{{\cal D}}_{-}}$ in the weak-star topology for all $s\geq
0$.

The paper is organized as follows. In Section 2.1 we introduce the Wick
rotation in the lapse and study its properties under foliation changing
diffeomorphism. The admissibility of the resulting invariantly defined complex
metrics is shown in Section 2.2. In Section 3.1 we collect some background
material on analytic semigroups and their generators, after which Section 3.2
addresses items (a)–(d) in the above list. The existence of a smooth kernel is
proven in Section 4, and Section 5 establishes its asymptotic expansion on the
diagonal. Finally, the strict Lorentzian limit is studied in Section 6, with
the result (g). We include three appendices: Appendix A on the (metric)
geometry of the foliations and the $1+d$ decomposition of diffeomorphisms.
Appendix B on the functional analytical arena: test functions, distributions
and Sobolev spaces. Appendix C provides a concise summary of a local
regularity theory adapted to our complex metrics, as needed for Section 4.

## 2\. Wick rotated lapse and admissible complex metrics

As outlined, we consider $1+d$ dimensional real, smooth manifolds $M$ that
admit a co-dimension-one foliation, $I\ni t\mapsto\Sigma_{t}$, see Appendix A.
In addition, $M$ is assumed to be equipped with a metric of the form

$ds^{2}_{\epsilon_{g}}=g^{\epsilon_{g}}_{\mu\nu}(y)dy^{\mu}dy^{\nu}=\epsilon_{g}N^{2}dt^{2}+{\rm
g}_{ab}(dx^{a}+N^{a}dt)(dx^{b}+N^{b}dt)\,,$ (2.1)

for both values of $\epsilon_{g}=\pm 1$. For both signatures, the leaves
$\Sigma_{t}$ of the foliation are the level sets of a smooth submersion
$T:M\to{\mathbb{R}}$ (referred to as a temporal function). When
$\epsilon_{g}=-1$, $dT$ is taken to be everywhere timelike and the (spacelike)
leaves are assumed to be Cauchy surfaces; the resulting Lorentzian manifolds
are globally hyperbolic. We are not aware of a concise established term for
the analogous $\epsilon_{g}=+1$ (Riemannian) manifolds. For short, we shall
refer to the metric components in (2.1) as the ADM (Arnowitt-Deser-Misner)
fields. These comprise a positive lapse $N>0$, the shift $N^{a}$, and the
positive definite spatial metric ${\rm g}_{ab}$. We collect these fields into
a triple $(N,N^{a},{\rm g}_{ab})_{\epsilon_{g}}$, where the temporal function
is tacit, and the subscript indicates the signature of the line element (2.1)
reconstructed from it.

For any foliation $I\ni t\mapsto\Sigma_{t}$ with associated ADM triple
$(N,N^{a},{\rm g}_{ab})_{\epsilon_{g}}$, our proposed notion of Wick rotation
is

$\mathfrak{w}_{\theta}:\quad(N,N^{a},{\rm
g}_{ab})_{\epsilon_{g}}\mapsto(i\epsilon_{g}^{-1/2}e^{-i\theta}N,N^{a},{\rm
g}_{ab})_{\epsilon_{g}},\quad\theta\in[0,\pi)\,,$ (2.2)

where $\sqrt{\epsilon_{g}}=+1,i$ for $\epsilon_{g}=1,-1$. This is such that,
starting from a fiducial foliation, one obtains a complexified line-element

$ds^{2}_{\epsilon_{g}}\mapsto ds^{2}_{\theta}=-e^{-2i\theta}N^{2}dt^{2}+{\rm
g}_{ab}(dx^{a}+N^{a}dt)(dx^{b}+N^{b}dt)\,.$ (2.3)

The case $\epsilon_{g}=-1$ gives $N\mapsto e^{-i\theta}N$ and relates a
Lorentzian signature ADM metric at $\theta=0$ to a complexified one that
becomes Euclidean for $\theta=\pi/2$. The case $\epsilon_{g}=+1$ gives
$N\mapsto ie^{-i\theta}N$ and relates the original Euclidean ADM metric at
$\theta=\pi/2$ to a complexified one that becomes Lorentzian for $\theta=0$.
The second half $(\pi/2,\pi)$ of the $\theta$ interval is carried along for
later use.

We write ${\rm Diff}(M)$ for the group of real diffeomorphisms
$U\ni(t,x)\mapsto(\chi^{0}(t,x),\chi^{a}(t,x))=(t^{\prime},{x^{\prime}}^{a})\in
U^{\prime}$ (for open neighborhoods $U,U^{\prime}$) that are smooth, connected
to the identity, as well as orientation preserving. An important subgroup
${\rm Diff}(\\{\Sigma\\})\subset{\rm Diff}(M)$ are the foliation preserving
diffeomorphisms of the form
$t^{\prime}=\chi^{0}(t),{x^{\prime}}^{a}=\chi^{a}(t,x)$. They preserve the
leaves $\Sigma_{t}$ of the foliation, potentially changing their time
labeling. The line elements (2.1) and (2.3) are manifestly invariant under
foliation preserving diffeomorphisms. In particular, the lapse Wick rotation
(2.2) does not depend on the choice of coordinates used to describe the given
fiducial foliation. A relevant question is, what happens if the foliation is
changed? To address this question we limit ourselves to foliations equivalent
to the original one, that is, foliations that can be reached by an actively
interpreted diffomorphism in ${\rm Diff}(M)$. An explicit formula for the
action of such foliation changing diffeomorphisms on the ADM data
$(N,N^{a},{\rm g}_{ab})_{\epsilon_{g}}$ will guide the analysis.

Another desirable property of a Wick rotation is to result in damping
integrands for a field theoretical functional integral. This leads to the
admissibility criterion for complex metrics proposed in [31, 29]. Here we
limit ourselves to a minimally coupled selfinteracting scalar field theory.
The signs in (2.2) are chosen such the resulting complex metric is admissible
in the chosen reference foliation. Again, the relevant issue is whether or not
admissibility is preserved in other foliations.

### 2.1 Foliation changing diffeomorphisms

The Wick rotation (2.2), (2.3) inevitably refers to a fiducial foliation. The
1-forms entering, i.e. $Ndt,e^{a}:=dx^{a}+N^{a}dt$, $a=1,\ldots,d$, comprise a
frame on $M$ which we dub the foliation frame. It is manifestly a coordinate
independent notion and thus invariant under passively interpreted
diffeomorphisms, as long as the foliation (i.e. the underlying temporal
function $T$) is held fixed. Upon transition to a different temporal function
$T^{\prime}$ whose level surfaces define a new (equivalent) foliation
$t^{\prime}\mapsto\Sigma_{t^{\prime}}$ the foliation frame transforms in a
nontrivial way. Writing
$(t^{\prime},{x^{\prime}}^{a})=(\chi^{0}(t,x),\chi^{a}(t,x))$ for the actively
interpreted diffeomorphisms, the transformation law comes out as

$\displaystyle N^{\prime}dt^{\prime}$
$\displaystyle=\frac{N}{D_{\epsilon_{g}}}\Big{[}Cdt+\frac{{\partial}t^{\prime}}{{\partial}x^{a}}e^{a}\Big{]}\,,$
$\displaystyle{e^{\prime}}^{a}$
$\displaystyle=X_{b}^{a}\Big{[}e^{b}-\epsilon_{g}{\rm
g}^{bc}\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\frac{N^{2}}{D_{\epsilon_{g}}^{2}}\Big{(}Cdt+\dfrac{{\partial}t^{\prime}}{{\partial}x^{d}}e^{d}\Big{)}\Big{]}\,,$
(2.4)

where

$\displaystyle D_{\epsilon_{g}}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\sqrt{C^{2}+\epsilon_{g}N^{2}\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}\dfrac{{\partial}t^{\prime}}{{\partial}x^{d}}{\rm
g}^{cd}}\,,\quad
C=\dfrac{{\partial}t^{\prime}}{{\partial}t}-\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}N^{c}\,,$
$\displaystyle X_{b}^{a}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{b}}-\frac{1}{C}\frac{{\partial}t^{\prime}}{{\partial}x^{b}}\Big{(}\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}t}-\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{d}}N^{d}\Big{)}\,.$
(2.5)

We refer to Appendix A for the block decomposition of diffeomorphisms; the
combinations (2.1) will occur frequently and always refer to a generic
underlying diffeomorphism that is suppressed in the notation. For the
derivation of (2.1) Appendix A of [38] may be consulted. The mathematical
equivalence between active and passive diffeomorphism transformations requires
that

$ds^{2}_{\epsilon_{g}}=\epsilon_{g}{N^{\prime}}^{2}{dt^{\prime}}^{2}+{\rm
g}^{\prime}_{ab}(d{x^{\prime}}^{a}+{N^{\prime}}^{a}dt^{\prime})(d{x^{\prime}}^{b}+{N^{\prime}}^{b}dt^{\prime})\,.$
(2.6)

This fixes the transformation law for ${\rm g}^{\prime}_{ab}$ and after
stripping off the coordinate 1-forms from $N^{\prime}dt^{\prime}$ and
${e^{\prime}}^{a}$ one one obtains the transformation law for the ADM triples
$(N,N^{a},{\rm g}_{ab})_{\epsilon_{g}}$ themselves [38]

${\rm transf}_{\epsilon_{g}}:(N,N^{2},{\rm g}_{ab})_{\epsilon_{g}}\
\longrightarrow\ (N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{\epsilon_{g}}\,,$ (2.7)

where

$\displaystyle N^{\prime}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{N}{D_{\epsilon_{g}}}$ (2.8a)
$\displaystyle{N^{\prime}}^{a}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{1}{D_{\epsilon_{g}}^{2}}\bigg{(}\Big{(}\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}t}-\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{d}}N^{d}\Big{)}C+\epsilon_{g}N^{2}\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{d}}\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}{\rm
g}^{cd}\bigg{)}$ (2.8b) $\displaystyle{{\rm g}^{\prime}}_{ab}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\Big{(}\dfrac{{\partial}x^{c}}{{\partial}{x^{\prime}}^{a}}+\frac{{\partial}t}{{\partial}{x^{\prime}}^{a}}N^{c}\Big{)}\Big{(}\dfrac{{\partial}x^{d}}{{\partial}{x^{\prime}}^{b}}+\frac{{\partial}t}{{\partial}{x^{\prime}}^{b}}N^{d}\Big{)}{\rm
g}_{cd}+\epsilon_{g}N^{2}\dfrac{{\partial}t}{{\partial}{x^{\prime}}^{a}}\dfrac{{\partial}t}{{\partial}{x^{\prime}}^{b}}\,.$
(2.8c)

Remarks.

(i) Upon linearization $t^{\prime}=t-\xi^{0}(t,x)+O((\xi^{0})^{2})$,
${x^{\prime}}^{a}=x^{a}-\xi^{a}(t,x)+O((\xi^{a})^{2})$,
$N^{\prime}=N+\delta_{\xi}N$, etc, the transformations (2.8) read

$\displaystyle\delta_{\xi}N$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\xi^{\mu}N_{\mu}N+N{\partial}_{t}\xi^{0}-NN^{a}{\partial}_{a}\xi^{0}\,,$
$\displaystyle\delta_{\xi}N^{a}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\xi^{\mu}{\partial}_{\mu}N^{a}+{\partial}_{t}\xi^{a}+N{\partial}_{t}\xi^{0}-N^{b}{\partial}_{b}\xi^{a}+{\partial}_{b}\xi^{0}(\epsilon_{g}N^{2}{\rm
g}^{ab}-N^{a}N^{b})\,,$ $\displaystyle\delta_{\xi}{\rm g}_{ab}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\xi^{0}{\partial}_{t}{\rm
g}_{ab}+N^{c}({\rm g}_{cb}{\partial}_{a}\xi^{0}+{\rm
g}_{ca}{\partial}_{b}\xi^{0})+{\cal L}_{\vec{\xi}}{\rm g}_{ab}\,.$ (2.9)

These generate the ‘group’ of infinitesimal Lagrangian gauge transformations
of a generally covariant system, c.f. [41]. Augmented by
$\delta_{\xi}\phi=\xi^{\mu}{\partial}_{\mu}\phi$, they comprise in particular
the gauge transformations of the scalar field action (2.26) below. Note that
the $\epsilon_{g}$ dependence now only enters in the $\delta_{\xi}N^{a}$ gauge
transformation. By analogy to (2.7) we shall write ${\rm
lintransf}_{\epsilon_{g}}(N,N^{a},{\rm
g}_{ab})_{\epsilon_{g}}=(\delta_{\xi}N,\delta_{\xi}N^{a},\delta_{\xi}{\rm
g}_{ab})_{\epsilon_{g}}$, with the understanding that the version of the
matching signature is used. Conversely, one should interpret (2.8) as the
finite gauge transformations characterizing a generally covariant system with
metrics in ADM form.

(ii) On the right hand sides of (2.8a), (2.8b) the new coordinates associated
to the temporal function $T^{\prime}=t^{\prime}$ occur as functions of the
original ones. In order to interpret the last relation in the same way the
inversion formulas (A) ought to be inserted. For readability’s sake we retain
the given expression (2.8c) as a shorthand.

(iii) The maps (2.8) are invertible, and the formulas for the inverse
transformations can be obtained simply by exchanging ‘primed’ with ‘unprimed’
quantities (fields and coordinate functions).

(iv) In addition to being highly nonlinear the transformation laws (2.1),
(2.8) also depend on the signature parameter. As in (2.1) this reflects the
fact that we take real, signature dependent metrics and the associated ADM
triples as a starting point. On triples $(\sqrt{\epsilon_{g}}N,N^{a},{\rm
g}_{ab})$ the foliation changing diffeomorphisms act in an $\epsilon_{g}$
independent way (formally given by the ${\rm transf}_{+}$ formulas).

(v) In the lapse transformation law a consistent square root needs to be
taken. This is possible since we restrict attention to separately time and
space orientation preserving diffeomorphisms. As far as the ADM metrics are
concerned one could work with triples $(N^{2},N^{a},{\rm
g}_{ab})_{\epsilon_{g}}$ where only the square of the lapse enters. Then ${\rm
transf}_{\epsilon_{g}}$ would act as in (2.8) just with $(N^{2})^{\prime}$
given by the square of the right hand side of (2.8a).

Using (2.1) one can deduce the transformation laws of covariant tensor
components defined with respect to the foliation frame. For example, for a co-
vector
$V_{\mu}dy^{\mu}=vNdt+v_{a}e^{a}=v^{\prime}N^{\prime}dt^{\prime}+v_{a}^{\prime}{e^{\prime}}^{a}$
one finds111The relations (2.10), (2.11) correct typos in the corresponding
formulas (A.53), (A.52) of [38].

$\displaystyle v^{\prime}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{D_{\epsilon_{g}}}\Big{(}Cv+\epsilon_{g}N\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}{\rm
g}^{cd}v_{d}\Big{)}\,,\quad
v^{\prime}_{a}=\Big{(}\frac{{\partial}x^{b}}{{\partial}{x^{\prime}}^{a}}+\frac{{\partial}t}{{\partial}{x^{\prime}}^{a}}N^{b}\Big{)}v_{b}+N\frac{{\partial}t}{{\partial}{x^{\prime}}^{a}}v\,.$
(2.10)

The frame dual to $(Ndt,e^{a})$ in the reference foliation consists of the
vector fields $(N^{-1}e_{0},{\partial}_{a})$. There are analogous
transformation formulas under a change of foliation, which can be found in
Appendix A of [38]. We shall only need the induced transformation formulas for
the components of a vector
$V^{\mu}{\partial}/{\partial}y^{\mu}=\epsilon_{g}\check{v}N^{-1}e_{0}+\check{v}^{a}{\partial}_{a}=\epsilon_{g}\check{v}^{\prime}{N^{\prime}}^{-1}e^{\prime}_{0}+\mbox{$\check{v}^{\prime}$}^{a}{\partial}^{\prime}_{a}$,
which read

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\check{v}^{\prime}\\!$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\\!\frac{1}{D_{\epsilon_{g}}}\Big{(}C\check{v}+\epsilon_{g}N\dfrac{{\partial}t^{\prime}}{{\partial}x^{a}}\check{v}^{a}\Big{)}\,,\;\;\check{v}^{\prime
a}=X^{a}_{b}\bigg{[}\check{v}^{b}-{\rm
g}^{bc}\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\frac{N}{D_{\epsilon_{g}}^{2}}\Big{(}C\check{v}+\epsilon_{g}N\dfrac{{\partial}t^{\prime}}{{\partial}x^{d}}\check{v}^{d}\Big{)}\bigg{]}.\,\,$
(2.11)

We now perform a Wick rotation (2.2) in the original foliation, resulting in
the complex metric (2.3). As in (2.2) we combine the complexified ADM fields
again into a triple $(N_{\theta}:=e^{-i\theta}N,N^{a},{\rm g}_{ab})_{-}$, with
the $-$ subscript indicating that the associated geometry arises through
(2.3), i.e. $ds_{\theta}^{2}=-N_{\theta}^{2}dt^{2}+\ldots$. Next, we subject
the fields $N_{\theta}:=e^{-i\theta}N,N^{a},{\rm g}_{ab}$ to a foliation
changing diffeomorphisms. The fields referring to the resulting equivalent
foliation $I\ni t^{\prime}\mapsto\Sigma^{\prime}_{t^{\prime}}$ are denoted by
a prime. On account of the sign convention in (2.3) we use the ${\rm
transf}_{-}$ transformations with its domain extended to allow for a complex
lapse. This gives $(N^{\prime}_{\theta},{N^{\prime}}_{\theta}^{a},{{\rm
g}^{\prime}}^{\theta}_{ab})_{-}={\rm transf}_{-}(N_{\theta},N^{a},{\rm
g}_{ab})_{-}$ with

$\displaystyle N_{\theta}^{\prime}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{N_{\theta}}{\sqrt{C^{2}-N_{\theta}^{2}\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}\dfrac{{\partial}t^{\prime}}{{\partial}x^{d}}{\rm
g}^{cd}}}\,,$ (2.12a) $\displaystyle{N^{\prime}_{\theta}}^{a}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\dfrac{\Big{(}\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}t}-\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{d}}N^{d}\Big{)}C-N_{\theta}^{2}\dfrac{{\partial}{x^{\prime}}^{a}}{{\partial}x^{d}}\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}{\rm
g}^{cd}}{C^{2}-N_{\theta}^{2}\dfrac{{\partial}t^{\prime}}{{\partial}x^{c}}\dfrac{{\partial}t^{\prime}}{{\partial}x^{d}}{\rm
g}^{cd}}\,,$ (2.12b) $\displaystyle{{\rm g}^{\prime}}^{\theta}_{ab}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\Big{(}\dfrac{{\partial}x^{c}}{{\partial}{x^{\prime}}^{a}}+\frac{{\partial}t}{{\partial}{x^{\prime}}^{a}}N^{c}\Big{)}\Big{(}\dfrac{{\partial}x^{d}}{{\partial}{x^{\prime}}^{b}}+\frac{{\partial}t}{{\partial}{x^{\prime}}^{b}}N^{d}\Big{)}{\rm
g}_{cd}-N_{\theta}^{2}\dfrac{{\partial}t}{{\partial}{x^{\prime}}^{a}}\dfrac{{\partial}t}{{\partial}{x^{\prime}}^{b}}\,.$
(2.12c)

The last relation should be interpreted in the same way as (2.8c).

The fact that also ${N^{\prime}_{\theta}}^{a}$, ${{{\rm
g}}^{\prime}}^{\theta}_{ab}$ are now complex in general highlights the sense
in which the Wick rotation (2.2) is foliation dependent. However, specializing
(2.12) to foliation preserving diffeomorphisms one sees that the $N_{\theta}$
dependence in ${N^{\prime}}^{a}$ and ${\rm g}^{\prime}_{ab}$ drops out, while
$N^{\prime}_{\theta}=e^{-i\theta}N^{\prime}=({\partial}t^{\prime}/{\partial}t)^{-1}N_{\theta}=({\partial}t^{\prime}/{\partial}t)^{-1}e^{-i\theta}N$
holds iff $N^{\prime}=({\partial}t^{\prime}/{\partial}t)^{-1}N$. Hence, the
definition (2.3) only depends on the foliation and not on the coordinatization
of the hypersurfaces or their time labels.

######  Proposition 2.1.

The lapse Wick rotated metric
$g^{\theta}_{\mu\nu}dy^{\mu}dy^{\nu}=-N_{\theta}^{2}dt^{2}+{\rm
g}_{ab}(dx^{a}+N^{a}dt)(dx^{b}+N^{b}dt)$ in the fiducial foliation
$t\mapsto\Sigma_{t}$ gives in a new (equivalent) foliation
$t^{\prime}\mapsto\Sigma^{\prime}_{t^{\prime}}$ rise to
${{g^{\prime}}^{\theta}}_{\\!\\!\\!\\!\mu\nu}d{y^{\prime}}^{\mu}d{y^{\prime}}^{\nu}=-{N^{\prime}_{\theta}}^{2}dt^{\prime
2}+{{\rm
g}^{\prime}}^{\theta}_{ab}(d{x^{\prime}}^{a}+{N^{\prime}_{\theta}}^{a}dt^{\prime})(d{x^{\prime}}^{b}+{N^{\prime}_{\theta}}^{b}dt^{\prime})$.
This is such that

$g^{\theta}_{\mu\nu}dy^{\mu}dy^{\nu}={{g^{\prime}}^{\theta}}_{\\!\\!\\!\\!\mu\nu}d{y^{\prime}}^{\mu}d{y^{\prime}}^{\nu}\,.$
(2.13)

We shall refer to (2.13) as the complexified metric defined by lapse Wick
rotation. It is invariantly defined with respect to passive and active
diffeomorphisms but depends on the choice of fiducial foliation.

Proof of Proposition 2.1. Viewing (2.12) as a definition only (2.13) needs to
be shown. This can be established by a a lengthy direct computation. ∎

Wick rotation in non-fiducial foliations. So far the Wick rotation (2.2) only
acted in the arbitrarily chosen but then fixed fiducial foliation. The result
was then transplanted to other foliations by a foliation changing
diffeomorphism. Formalizing this construction, one can define a Wick rotation
in a non-fiducial foliation by the alternative expressions

$\displaystyle\mathfrak{w}^{\prime}_{\theta}$ $\displaystyle:=$
$\displaystyle{\rm transf}_{-}\circ\mathfrak{w}_{\theta}\circ({\rm
transf}_{-})^{-1}\,,$ $\displaystyle\mathfrak{w}^{\prime}_{\theta}$
$\displaystyle:=$ $\displaystyle{\rm
transf}_{+}\circ\mathfrak{w}_{\theta}\circ({\rm transf}_{+})^{-1}\,.$ (2.14)

Here, $\mathfrak{w}^{\prime}_{\theta}$ acts on the real triples
$(N^{\prime},{N^{\prime}}^{a},{\rm g}^{\prime}_{ab})_{-}$ and
$(N^{\prime},{N^{\prime}}^{a},{\rm g}^{\prime}_{ab})_{+}$, respectively, of a
matching signature metric in a non-fiducial foliation. In the second ${\rm
transf}_{\epsilon_{g}}$ map its action is extended to allows for a complex
lapse. In the notation (2.12) the result is
$\mathfrak{w}^{\prime}_{\theta}(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{-}=(N^{\prime}_{\theta},{N^{\prime}}_{\theta}^{a},{{\rm
g}^{\prime}}^{\theta}_{ab})_{-}$ and
$\mathfrak{w}^{\prime}_{\theta}(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{+}=(iN^{\prime}_{\theta},{N^{\prime}}_{\theta}^{a},{{\rm
g}^{\prime}}^{\theta}_{ab})_{+}$. Since $(iN^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{+}=(N^{\prime},{N^{\prime}}^{a},{\rm g}^{\prime}_{ab})_{-}$
and $(iN^{\prime}_{\theta},{N^{\prime}}_{\theta}^{a},{{\rm
g}^{\prime}}^{\theta}_{ab})_{+}=(N^{\prime}_{\theta},{N^{\prime}}_{\theta}^{a},{{\rm
g}^{\prime}}^{\theta}_{ab})_{-}$, both variants of (2.1) are consistent; we
keep both so as to be able to work with real (signature dependent) triples
before Wick rotation.

The cases $\theta=\pi/2,0$ are of particular interest and define a Wick flip.
Specializing the defining relations in the fiducial foliation
$\mathfrak{w}_{\theta}(N,N^{a},{\rm g}_{ab})_{-}=(e^{-i\theta}N,N^{a},{\rm
g}_{ab})_{-}$, $\mathfrak{w}_{\theta}(N,N^{a},{\rm
g}_{ab})_{+}=(ie^{-i\theta}N,N^{a},{\rm g}_{ab})_{+}$, to these cases one has

$\displaystyle\mathfrak{w}_{\pi/2}(N,N^{a},{\rm g}_{ab})_{-}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle(-iN,N^{a},{\rm
g}_{ab})_{-}=(N,N^{a},{\rm g}_{ab})_{+}\,,$
$\displaystyle\mathfrak{w}_{\pi/2}(N,N^{a},{\rm g}_{ab})_{+}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle(N,N^{a},{\rm g}_{ab})_{+}\,,$
$\displaystyle\mathfrak{w}_{0}(N,N^{a},{\rm g}_{ab})_{-}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle(N,N^{a},{\rm g}_{ab})_{-}\,,$
$\displaystyle\mathfrak{w}_{0}(N,N^{a},{\rm g}_{ab})_{+}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle(iN,N^{a},{\rm
g}_{ab})_{+}=(N,N^{a},{\rm g}_{ab})_{-}\,.$ (2.15)

Note that $\mathfrak{w}_{\pi/2}^{2}=\mathfrak{w}_{\pi/2}$,
$\mathfrak{w}_{0}^{2}=\mathfrak{w}_{0}$, and
$\mathfrak{w}_{0}\mathfrak{w}_{\pi/2}=\mathfrak{w}_{0}$,
$\mathfrak{w}_{\pi/2}\mathfrak{w}_{0}=\mathfrak{w}_{\pi/2}$. Clearly, the
${\rm transf}_{+}$ version of (2.1) is trivial for $\mathfrak{w}_{\pi/2}$
while the ${\rm transf}_{-}$ version of (2.1) is trivial for
$\mathfrak{w}_{0}$. The other two relations are

$\displaystyle\mathfrak{w}^{\prime}_{\pi/2}$ $\displaystyle:=$
$\displaystyle{\rm transf}_{-}\circ\mathfrak{w}_{\pi/2}\circ({\rm
transf}_{-})^{-1}\,,$ $\displaystyle\mathfrak{w}^{\prime}_{0}$
$\displaystyle:=$ $\displaystyle{\rm
transf}_{+}\circ\mathfrak{w}_{0}\circ({\rm transf}_{+})^{-1}\,,$ (2.16)

and extend the Wick flip to non-fiducial foliations. Explicitly,
$\mathfrak{w}^{\prime}_{\pi/2}(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{-}=(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{+}$, and
$\mathfrak{w}^{\prime}_{0}(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{+}=(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{-}$.

Complexified metric as a rank one perturbation. In the fiducial foliation the
complexified metric can trivially be interpreted as a rank one deformation of
the original one. Writing, in adapted coordinates,
$g^{(\epsilon_{g})}_{\mu\nu}dy^{\mu}dy^{\nu}=\epsilon_{g}N^{2}dt^{2}+{\rm
g}_{ab}e^{a}e^{b}$ and
$g^{\theta}_{\mu\nu}dy^{\mu}dy^{\nu}=-N_{\theta}^{2}dt^{2}+{\rm
g}_{ab}e^{a}e^{b}$ one has

$g^{\theta}_{\mu\nu}dy^{\mu}dy^{\nu}=g^{(\epsilon_{g})}_{\mu\nu}dy^{\mu}dy^{\nu}-(\epsilon_{g}+e^{-2i\theta})N^{2}dTdT\,,$
(2.17)

with $dT$ the differential of the temporal function of the foliation. In other
(primed) coordinates associated with another temporal function $T^{\prime}$,
we seek to compare
${g^{\prime}}^{\theta}_{\\!\mu\nu}{dy^{\prime}}^{\mu}{dy^{\prime}}^{\nu}$ as
in Proposition 2.1 with
${g^{\prime}}^{(\epsilon_{g})}_{\mu\nu}{dy^{\prime}}^{\mu}{dy^{\prime}}^{\nu}$
from the right hand side of (2.6). One might guess that the deformation term
in the new foliation arises simply by placing ‘appropriate primes’ on the
original deformation, i.e. $N^{\prime 2}dT^{\prime}dT^{\prime}$. However, this
is not the case, the correct assertion being

######  Proposition 2.2.

The lapse Wick rotated metric, defined with respect to a fiducial foliation in
(2.2), is a rank one perturbation with a metric dependent covector field. In
any foliation equivalent to the fiducial one,

${g^{\prime}}^{\theta}_{\mu\nu}={g^{\prime}}^{(\epsilon_{g})}_{\mu\nu}-(\epsilon_{g}+e^{-2i\theta})\big{(}v^{\prime}N^{\prime}{\partial}^{\prime}_{\mu}t^{\prime}+v_{a}^{\prime}{e^{\prime}}^{a}_{\mu}\big{)}\big{(}v^{\prime}N^{\prime}{\partial}^{\prime}_{\nu}t^{\prime}+v_{a}^{\prime}{e^{\prime}}^{a}_{\nu}\big{)}\,,$
(2.18)

where with the notation from (2.1)

$v^{\prime}=\frac{C}{D_{\epsilon_{g}}}\,,\quad
v_{a}^{\prime}=N\frac{{\partial}t}{{\partial}{x^{\prime}}^{a}}\,.$ (2.19)

Here, the ${\partial}t/{\partial}{x^{\prime}}^{a}$ term should again be
interpreted in terms of $t^{\prime}$ via the inversion formula in (A).

Proof. The origin of the expressions for $(v^{\prime},v_{a}^{\prime})$ is
simply as the image of
$vNdt+v_{a}e^{a}=v^{\prime}N^{\prime}dt^{\prime}+v_{a}^{\prime}{e^{\prime}}^{a}$
for $v=1,v_{a}=0$, using (2.10). The last identity reaffirms the mathematical
equivalence between passive and active diffeomorphism transformations, for the
perturbing covector field. Since the latter is already known to hold for the
unperturbed metric via (2.6) and the Wick rotated one via Prop. 2.1 it follows
that

$\displaystyle-{N^{\prime}_{\theta}}^{2}dt^{\prime 2}+{{\rm
g}^{\prime}}^{\theta}_{ab}(d{x^{\prime}}^{a}+{N^{\prime}_{\theta}}^{a}dt^{\prime})(d{x^{\prime}}^{b}+{N^{\prime}_{\theta}}^{b}dt^{\prime})$
$\displaystyle\quad=\epsilon_{g}{N^{\prime}}^{2}d{t^{\prime}}^{2}+{\rm
g}^{\prime}_{ab}{e^{\prime}}^{a}{e^{\prime}}^{b}-(\epsilon_{g}+e^{-2i\theta})(v^{\prime}N^{\prime}dt^{\prime}+v_{a}^{\prime}{e^{\prime}}^{a})^{2}\,.$
(2.20)

Upon stripping off the coordinate differentials $d{y^{\prime}}^{\mu}$ one
obtains (2.18). ∎

Remarks.

(i) The identity (2.1) can also be verified by a lengthy direct computation,
using the formulae from Appendix A of [38]. Note that the phase $e^{-i\theta}$
enters the defining relations (2.12) highly nonlinearly on the left hand side
while appears only quadratically on the right hand side.

(ii) A notion of Wick rotation by a rank one deformation with a complex
coefficient $\lambda$ has first been proposed in [6]. Their perturbing
covector field $V_{\mu}$ is, however, taken as a metric independent additional
structure on the manifold. For non-extreme values of $\lambda$ the perturbed
metric and all concepts derived from it will depend on the choice of
$V_{\mu}$. In the present setting the perturbing covector is itself defined in
terms of the metric data. Our complexified metric analogously depends on the
choice of fiducial foliation.

(iii) In [43, 29, 46] the complexification is done in the internal metric of a
Vielbein basis. That is, the Vielbein is kept real and merely the scalar
diagonal coefficients are replaced by phases. In the present foliated setting
the natural Vielbein for (2.1) is

$\displaystyle E_{I}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
N^{-1}e_{0}\,\epsilon_{I}+\epsilon_{I}^{a}{\partial}_{a}=E_{I}^{\mu}\frac{{\partial}}{{\partial}y^{\mu}}\,,$
$\displaystyle E^{I}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\epsilon_{g}Ndt\,\epsilon^{I}+\epsilon^{I}_{a}e^{a}=E^{I}_{\mu}dy^{\mu}\,,$
(2.21)

where $E_{I}^{\mu}E_{\mu}^{J}=\delta_{I}^{J}$,
$E_{I}^{\mu}E_{\nu}^{I}=\delta^{\mu}_{\nu}$, $I,J=0,\ldots,d$, and
$g^{(\epsilon_{g})}_{\mu\nu}dy^{\mu}dy^{\nu}=\delta_{IJ}$ expresses the
desired complete diagonalization. The defining relations for the component
fields $(\epsilon_{I},\epsilon_{I}^{a})$ and $(\epsilon^{I},\epsilon^{I}_{a})$
can be read off upon inserting (A). Applying the lapse Wick rotation (2.2) to
(2.1) would preserve the strict diagonalization at the expense of
complexifying the Vielbeins. A better option is to retain the real Vielbeins
(2.1) and use the rank one formula (2.17) to infer

$g^{\theta}_{\mu\nu}E_{I}^{\mu}E_{J}^{\nu}=\delta_{IJ}-(\epsilon_{g}+e^{-2i\theta})\epsilon_{I}\epsilon_{J}\,.$
(2.22)

This is no longer fully diagonal but has eigenvalues
$(-e^{-2i\theta},1,\ldots,1)$. The transformation formulas (2.1) can be used
to deduce the induced behavior of the $\epsilon^{I},\epsilon^{I}_{a}$ under
foliation changing diffeomorphisms, and similarly for
$\epsilon_{I},\epsilon_{I}^{a}$. This retains the covariance in a sense
analogous to the rank one perturbations (2.18).

(iv) For later use we also prepare the counterpart of the rank one deformation
formula (2.1), (2.18) for the inverse metric. In the fiducial foliation one
has

$g_{\theta}^{\mu\nu}(y)\frac{{\partial}}{{\partial}y^{\mu}}\frac{{\partial}}{{\partial}y^{\nu}}=g_{\epsilon_{g}}^{\mu\nu}(y)\frac{{\partial}}{{\partial}y^{\mu}}\frac{{\partial}}{{\partial}y^{\nu}}-(\epsilon_{g}+e^{+2i\theta})N^{-2}e_{0}^{2}\,.$
(2.23)

The image in a generic foliation can be found in parallel to (2.18), (2.19)
using (2.11). for $\check{v}=1,\check{v}^{a}=0$.

${g^{\prime}}_{\\!\\!\theta}^{\mu\nu}(y^{\prime})\frac{{\partial}}{{\partial}{y^{\prime}}^{\mu}}\frac{{\partial}}{{\partial}{y^{\prime}}^{\nu}}={g^{\prime}}_{\\!\\!\epsilon_{g}}^{\mu\nu}(y^{\prime})\frac{{\partial}}{{\partial}{y^{\prime}}^{\mu}}\frac{{\partial}}{{\partial}{y^{\prime}}^{\nu}}-(\epsilon_{g}+e^{+2i\theta})\big{(}\epsilon_{g}\check{v}^{\prime}{N^{\prime}}^{-1}e^{\prime}_{0}+\mbox{$\check{v}^{\prime}$}^{a}\,{\partial}^{\prime}_{a}\big{)}^{2}\,,$
(2.24)

where

$\check{v}^{\prime}=\frac{C}{D_{\epsilon_{g}}}\,,\quad\mbox{$\check{v}^{\prime}$}^{a}=-\frac{NC}{D_{\epsilon_{g}}^{2}}X_{b}^{a}{\rm
g}^{bc}\frac{{\partial}t^{\prime}}{{\partial}x^{c}}\,.$ (2.25)

### 2.2 Admissibility criteria for scalar field theories.

One reasonable “admissibility criterion” for a complex metric
$g^{\theta}_{\mu\nu}dy^{\mu}dy^{\nu}$ on a real manifold is that the
classically interpreted exponential of the action entering the functional
integral is damping. This reasoning is tacit in numerous discussions of Wick
rotations, recent explicit accounts are [31, 29, 46]. Taking Lorentzian
signature as basic and writing $S_{\theta}=S_{-}|_{g\mapsto g^{\theta}}$ for
the complexified action, $e^{iS_{\theta}}$ should be damping. That is, ${\rm
Im}S_{\theta}>0$, for some range of $\theta>0$, if $S_{\theta=0}=S_{-}$ is the
Lorentzian signature action. For short, we call a complex metric
$g^{\theta}_{\mu\nu}dy^{\mu}dy^{\nu}$ on a real manifold admissible if this
condition is met for the field theories under consideration. In a small
$\theta$ expansion the linear response, $S_{\theta}=S_{-}+(\delta S_{-}/\delta
g_{\mu\nu})(g^{\theta}-g)_{\mu\nu}+O(\theta^{2})$, relates to the energy-
momentum tensor $T_{-}^{\mu\nu}=-(2/\sqrt{g})\delta S_{-}/\delta g_{\mu\nu}$,
of the Lorentzian theory. The condition ${\rm Im}S_{\theta}>0$ is then to
$O(\theta)$ typically satisfied if the energy momentum tensor satisfies the
weak energy condition (WEC). For short, we call a complex metric WEC
admissible if $\imaginary S_{\theta}>0$ holds to $O(\theta)$ on account of the
WEC condition.

On a foliated manifold both criteria are manifestly coordinate independent
(invariant under passive diffeomorphisms) as long as the fiducial foliation is
kept fixed. Below we limit ourselves to self-interacting scalar fields on a
foliated background and address the admissibility of our lapse Wick rotated
complexified metric in foliations other than the fiducial one in which the
rotation is defined. Somewhat surprisingly, the analysis is conceptually
different for the exact Wick rotation and the version linearized in $\theta$.

We prepare the scalar field action for both signatures

$\displaystyle S_{\epsilon_{g}}[\phi,g]$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\epsilon_{g}\\!\int\\!dy\sqrt{\epsilon_{g}g}\Big{\\{}\frac{1}{2}g_{\epsilon_{g}}^{\mu\nu}{\partial}_{\mu}\phi{\partial}_{\nu}\phi+U(\phi)\Big{\\}}$
(2.26) $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\int\\!dt\\!\int_{\Sigma}\\!d^{d}x\sqrt{{\rm
g}}\Big{\\{}\frac{1}{2N}e_{0}(\phi)^{2}+\frac{\epsilon_{g}}{2}N{\rm
g}^{ab}{\partial}_{a}\phi{\partial}_{b}\phi+\epsilon_{g}NU(\phi)\Big{\\}}\,.$

In the second line we display the $1\\!+\\!d$ form of the action in some
fiducial foliation with metric data $(N,N^{a},{\rm g}_{ab})_{\epsilon_{g}}$.
Further, $U(\phi)$ is a metric independent potential which we assume to be
non-negative. The bi-transversal component of the energy momentum tensor
$T^{\epsilon_{g}}_{\mu\nu}$ is defined by projection with a real vector
$m^{\mu}$ satisfying $dt_{\mu}m^{\mu}=1$,
$m^{\mu}m^{\nu}g^{\epsilon_{g}}_{\mu\nu}=\epsilon_{g}N^{2}$. This gives

$\displaystyle T^{\epsilon_{g}}_{\mu\nu}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{2\epsilon_{g}}{\sqrt{g}}\frac{\delta
S_{\epsilon_{g}}}{\delta
g^{\mu\nu}}={\partial}_{\mu}\phi{\partial}_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}{\partial}_{\rho}\phi{\partial}_{\sigma}\phi-
g_{\mu\nu}U(\phi)\,,$ $\displaystyle
N^{-2}m^{\mu}m^{\nu}T^{\epsilon_{g}}_{\mu\nu}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{2N^{2}}e_{0}(\phi)^{2}-\frac{\epsilon_{g}}{2}{\rm
g}^{ab}{\partial}_{a}\phi{\partial}_{b}\phi-\epsilon_{g}U(\phi)\,,$ (2.27)

where we momentarily omit the $\epsilon_{g}$ sub/superscripts on the metric
for readability’s sake. One sees that $m^{\mu}m^{\nu}T^{-}_{\mu\nu}\geq 0$, so
Lorentzian signature scalar field theories with a non-negative potential
satisfy the WEC.

The action $S_{\epsilon_{g}}$ is manifestly invariant under foliation
preserving diffeomorphisms. In fact, each of the terms
$N^{-2}e_{0}(\phi)^{2}$, ${\rm g}^{ab}{\partial}_{a}\phi{\partial}_{b}\phi$,
$U(\phi)$ is separately a scalar under ${\rm Diff}(\\{\Sigma\\})$ and the Wick
rotation (2.2) can unambiguously be applied. Explicitly, we define in the
fiducial foliation the lapse Wick rotated action by

$\displaystyle S_{\theta}[\phi,g]$ $\displaystyle:=$ $\displaystyle
S_{-}[\phi,g]\big{|}_{N\mapsto e^{-i\theta}N}=iS_{+}[\phi,g]\big{|}_{N\mapsto
ie^{-i\theta}N}$ (2.28) $\displaystyle\\!\\!=\\!\\!$ $\displaystyle\cos\theta
S_{-}[\phi,g]+i\sin\theta S_{+}[\phi,g]\,,$

where $S_{\pm}$ are given by the second line in (2.26). For $\theta\in(0,\pi)$
one has $\imaginary[S_{\theta}]>0$ and the generalized Boltzmann factor
$e^{+iS_{\theta}}$ in a functional integral is damping. It is thus plain that
the complexified action (2.28) is admissible in the above sense in the
fiducial foliation. To linear order, $S_{\theta}=S_{-}+i\theta
S_{+}+O(\theta^{2})$. Consistency with the WEC criterion requires that

$\displaystyle S_{+}$
$\displaystyle\stackrel{{\scriptstyle\displaystyle{!}}}{{=}}$
$\displaystyle\lim_{\theta\rightarrow
0}\frac{1}{i\theta}\int\\!\\!dtd^{d}x\,\frac{\delta S_{-}}{\delta
g^{\mu\nu}}(g_{-})\big{(}g^{\theta}-g_{-}\big{)}^{\mu\nu}$ (2.29)
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\lim_{\theta\rightarrow
0}\frac{e^{2i\theta}-1}{2i\theta}\int\\!\\!dtd^{d}x\,\sqrt{-g_{-}}N^{-2}\,T^{-}_{\mu\nu}m^{\mu}m^{\nu}\geq
0\,,$

where we used (2.23) and the variational definition of the energy momentum
tensor. Inserting (2.2) this is indeed an identity.

WEC admissibility in non-fiducial foliations. The fiducial foliation can of
course be chosen arbitrarily and in this sense (2.29) holds in any foliation
with its associated temporal function $T$. One can, however, also ask if
(2.29) continues to hold if the foliation is changed via the transformations
(2.8). From the mathematical equivalence between active and passive
diffeomorphism transformations one expects $T^{-}_{\mu\nu}m^{\mu}m^{\nu}$ not
to be invariant (being the time-time component of a ${0\choose 2}$ tensor) and
the issue is whether it remains positive. By comparing the second lines of
(2.26) and (2.2) one sees that both $S_{+}$ and $T^{-}_{\mu\nu}m^{\mu}m^{\nu}$
contain the sum of the temporal and the spatial gradient terms. By extension
of Proposition 2.1 these sums are scalars under ${\rm transf}_{+}$ in (2.8).
However, $T^{-}_{\mu\nu}$, stemming from the Lorentzian action should really
be subjected to the ${\rm transf}_{-}$ transformations, and will then not be a
scalar.

It is instructive to compute explicitly the transformation law of the sum and
difference of the temporal and the spatial gradient parts in the action
$S_{\epsilon_{g}}$ based on the matching ${\rm transf}_{\epsilon_{g}}$ version
of the transition formulas. Using the results from Appendix A of [38] one
finds

$\displaystyle[{N^{\prime}}^{-1}e_{0}^{\prime}(\phi^{\prime})]^{2}+\epsilon_{g}{{\rm
g}^{\prime}}^{ab}{\partial}^{\prime}_{a}\phi^{\prime}{\partial}^{\prime}_{b}\phi^{\prime}=[N^{-1}e_{0}(\phi)]^{2}+\epsilon_{g}{{\rm
g}}^{ab}{\partial}_{a}\phi{\partial}_{b}\phi\,,$ (2.30a)
$\displaystyle[{N^{\prime}}^{-1}e_{0}^{\prime}(\phi^{\prime})]^{2}-\epsilon_{g}{{\rm
g}^{\prime}}^{ab}{\partial}^{\prime}_{a}\phi^{\prime}{\partial}^{\prime}_{b}\phi^{\prime}=\dfrac{1}{D_{\epsilon_{g}}^{2}}\bigg{\\{}\Big{[}\frac{C}{N}e_{0}(\phi)+\epsilon_{g}N\frac{{\partial}t^{\prime}}{{\partial}x^{c}}{\rm
g}^{cd}{\partial}_{d}\phi\Big{]}^{2}$ $\displaystyle\makebox[28.45274pt]{
}-\epsilon_{g}{\rm
g}^{cd}\Big{[}C{\partial}_{c}\phi-\frac{{\partial}t^{\prime}}{{\partial}x^{c}}e_{0}(\phi)\Big{]}\Big{[}C{\partial}_{d}\phi-\frac{{\partial}t^{\prime}}{{\partial}x^{d}}e_{0}(\phi)\Big{]}\bigg{\\}}.$
(2.30b)

The first combination occurs in the Lagrangian of $S_{\epsilon_{g}}$ and
(2.30a) confirms the expected scalar transformation law. The sign flipped
version occurs in the bi-transversal component of the energy momentum tensor
(2.2) and, as expected, does not transform as a scalar under ${\rm
transf}_{\epsilon_{g}}$. Relevant in the present context is that the right
hand side of (2.30) can be written so that for $\epsilon_{g}=-1$ is is
manifestly non-negative. Hence, when subjecting the second line of (2.29) to
an active foliation changing diffeomorphism of the inherited signature type,
${\rm transf}_{-}$, its value changes but it remains positive.

Admissibility in non-fiducial foliations. The reason for slightly belaboring
the above point is that the situation is conceptually different if the
dependence on the phase $e^{\pm i\theta}$ is treated exactly and no reference
to the energy momentum tensor of the original Lorentzian action is made. To
frame the discussion it is convenient to define
$L(\phi,A):=\frac{1}{2}A^{\mu\nu}{\partial}_{\mu}\phi{\partial}_{\nu}\phi+U(\phi)$,
for any complex maximal rank matrix $A^{\mu\nu}$. Then, in a given fiducial
foliation $L(\phi,g_{+})$ is the Euclidean signature Lagrangian,
$-L(\phi,g_{-})$ is the Lorentzian signature Lagrangian, and
$-L(\phi,g_{\theta})$ is the Lagrangian of the complexified action (2.28),
excluding the complexified measure term $\sqrt{-g_{\theta}}$. We interpret
this measure term as $\sqrt{-g_{\theta}}=e^{-i\theta}\sqrt{\mp
g_{\mp}}=e^{-i\theta}N\sqrt{{\rm g}}$. Taking the extra phase into account the
Lagrangian of the complexified action with the real $N\sqrt{{\rm g}}$ measure
is $L_{\theta}=-e^{-i\theta}L(\phi,g_{\theta})$. In this notation the relation
(2.28) reads

$-e^{-i\theta}L(\phi,g_{\theta})(y)=-\cos\theta\,L(\phi,g_{-})(y)+i\sin\theta\,L(\phi,g_{+})(y)\,,$
(2.31)

where $y^{\mu}=(t,x^{a})$ are local coordinates adapted to the fiducial
foliation. The interplay with non-fidual foliations is described by

######  Proposition 2.3.

The Lagrangian $-e^{-i\theta}L(\phi,g_{\theta})(y)$ of the complexified action
is a scalar under the transformations (2.12),
$-e^{-i\theta}L(\phi,g_{\theta})(y)=-e^{-i\theta}L(\phi^{\prime},g_{\theta}^{\prime})(y^{\prime})$.
Explicitly,

$\frac{1}{2N_{\theta}^{2}}e_{0}(\phi)^{2}-\frac{1}{2}{\rm
g}^{ab}{\partial}_{a}\phi{\partial}_{b}\phi-U(\phi)=\frac{1}{2{N_{\theta}^{\prime}}^{2}}e^{\prime}_{0}(\phi^{\prime})^{2}-\frac{1}{2}{{\rm
g}^{\prime}}_{\theta}^{ab}{\partial}^{\prime}_{a}\phi^{\prime}{\partial}^{\prime}_{b}\phi^{\prime}-U(\phi^{\prime})\,,$
(2.32)

where
$e_{0}^{\prime}={\partial}_{t}^{\prime}-{N^{\prime}}^{a}{\partial}_{a}^{\prime}$
and ${{\rm g}^{\prime}}_{\theta}^{ab}$ is the inverse of ${{\rm
g}^{\prime}}^{\theta}_{ab}$ in (2.12c). Further,

$-e^{-i\theta}L(\phi^{\prime},g^{\prime}_{\theta})(y^{\prime})=-\cos\theta\,L(\phi^{\prime},g^{\prime}_{-})(y^{\prime})+i\sin\theta\,L(\phi^{\prime},g^{\prime}_{+})(y^{\prime})\,.$
(2.33)

In particular, the real and imaginary parts of
$-e^{-i\theta}L(\phi,g_{\theta})$ are separately scalars under the
transformations (2.12).

Proof. Since the inverse of the complexified metric enters the ‘covariant’
form of the action $S^{\theta}[\phi,g]=S_{-}[\phi,g^{\theta}]$ the assertion
(2.32) does not quite follow from (2.13). However, defining the inverses
$g_{\theta}^{\mu\nu}$ of $g^{\theta}_{\mu\nu}$ and
${g^{\prime}}_{\theta}^{\mu\nu}$ of ${g^{\prime}}^{\theta}_{\mu\nu}$ in the
obvious way with respect to the real vector field bases
${\partial}/{\partial}y^{\mu}$ and ${\partial}/{\partial}{y^{\prime}}^{\mu}$,
respectively, it is clear that

$g_{\theta}^{\mu\nu}\frac{{\partial}}{{\partial}y^{\mu}}\frac{{\partial}}{{\partial}y^{\mu}}={g_{\theta}^{\prime}}^{\mu\nu}\frac{{\partial}}{{\partial}{y^{\prime}}^{\mu}}\frac{{\partial}}{{\partial}{y^{\prime}}^{\mu}}\,,$
(2.34)

will hold as well. This implies (2.32).

The phase $e^{-i\theta}$ occurs highly nonlinearly on the right hand side of
(2.32). It is thus not immediate that the latter can be decomposed as claimed
on the right hand side of (2.33). To see that this is the case, we return to
(2.24) and insert it into the left hand side of (2.33). In a first step this
gives

$\displaystyle-e^{-i\theta}L(\phi^{\prime},g^{\prime}_{\theta})(y^{\prime})=e^{-i\theta}\Big{\\{}\\!-\frac{1}{2}{g^{\prime}_{\epsilon_{g}}}^{\\!\\!\mu\nu}{\partial}^{\prime}_{\mu}\phi^{\prime}{\partial}^{\prime}_{\nu}\phi^{\prime}-U(\phi^{\prime})$
$\displaystyle\quad-\frac{1}{2}(\epsilon_{g}+e^{+2i\theta})\big{(}\epsilon_{g}\check{v}^{\prime}{N^{\prime}}^{-1}e^{\prime}_{0}(\phi^{\prime})+\mbox{$\check{v}^{\prime}$}^{a}\,{\partial}^{\prime}_{a}\phi^{\prime}\big{)}^{2}\Big{\\}}\,.$
(2.35)

By construction, either sign $\epsilon_{g}=\pm 1$ can be chosen to evaluate
the right hand side. Choosing $\epsilon_{g}=+1$ one finds

$\displaystyle-e^{-i\theta}L(\phi^{\prime},g^{\prime}_{\theta})(y^{\prime})=i\sin\theta\Big{\\{}\frac{1}{2}{g^{\prime}_{+}}^{\\!\\!\mu\nu}{\partial}^{\prime}_{\mu}\phi^{\prime}{\partial}^{\prime}_{\nu}\phi^{\prime}+U(\phi^{\prime})\Big{\\}}$
$\displaystyle\quad-\cos\theta\,\Big{\\{}\frac{1}{2}{g^{\prime}_{+}}^{\\!\\!\mu\nu}{\partial}^{\prime}_{\mu}\phi^{\prime}{\partial}^{\prime}_{\nu}\phi^{\prime}-\big{(}\epsilon_{g}\check{v}^{\prime}{N^{\prime}}^{-1}e^{\prime}_{0}(\phi^{\prime})+\mbox{$\check{v}^{\prime}$}^{a}\,{\partial}^{\prime}_{a}\phi^{\prime}\big{)}^{2}-U(\phi^{\prime})\Big{\\}}\,.$
(2.36)

The first two terms in the second curly bracket can be simplified using the
$\theta=0$, $\epsilon_{g}=+1$ version of (2.24) in reverse. This yields
(2.33). ∎

In the context of our previous discussion of the WEC admissibility, the result
(2.33) is somewhat surprising. While in (2.29) the imaginary part of the
$O(\theta)$ perturbation is not a scalar under the inherited ${\rm
transf}_{-}$ transformation, the real and the imaginary parts in (2.33)
suddenly are. This is because the complex transformations (2.12) automatically
apply the matching transformations ${\rm trans}_{\pm}$ to the definite
signature parts of the quantities occurring on the right hand side of (2.24).
As a consequence, after reexpressing
$-e^{-i\theta}L(\phi^{\prime},g^{\prime}_{\theta})(y^{\prime})$ in terms of
the definite signature $L(\phi^{\prime},g^{\prime}_{-})$ and
$L(\phi^{\prime},g^{\prime}_{+})$ the latter coincide with the images of
$L(\phi,g_{-})$ and $L(\phi,g_{+})$ under the matching ${\rm transf}_{-}$ and
${\rm transf}_{+}$ transformations, respectively. There is no inherited
transformation law that is kept fixed and results in a non-scalar
transformation law.

Complexified Hessian. Next, we consider the Hessian defined by the quadratic
part of the action $S_{\theta}$. The appropriate background-fluctuation split
is $\phi=\varphi+f$, for a background $\varphi$ and some $f\in
C_{c}^{\infty}(M)$. Expanding the action to quadratic order in $f$ one has

$S_{\theta}[\varphi+f,g]=S_{\theta}[\varphi,g]-\int\\!\\!dt\\!\int_{\Sigma}\\!d^{d}x\,N\sqrt{{\rm
g}}\,f\,i\Delta_{\theta}\varphi-\frac{1}{2}\int\\!\\!dt\\!\int_{\Sigma}\\!d^{d}x\,N\sqrt{{\rm
g}}f\,i\Delta_{\theta}f+O(f^{3})\,.$ (2.37)

The Hessian $-i\Delta_{\theta}$ can be written in several alternatively useful
ways

$\displaystyle-i\Delta_{\theta}$
$\displaystyle=-e^{-i\theta}\big{[}-\nabla_{-}^{2}\big{|}_{N\mapsto
e^{-i\theta}N}+V\big{]}=-e^{i\theta}\nabla_{t}^{2}+e^{-i\theta}\nabla_{s}^{2}-e^{-i\theta}V$
$\displaystyle=-\cos\theta\,{\cal D}_{-}+i\sin\theta\,{\cal D}_{+}\,.$ (2.38)

We do not require $\varphi$ to be a solution of $\Delta_{\theta}\varphi=0$ and
treat the potential $V=U^{\prime\prime}(\varphi)$ as a given scalar function
on $M$. Further, ${\cal D}_{\pm}:=-\nabla_{\pm}^{2}+V$ are the
Euclidean/Lorentzian signature Hessians, respectively. The corresponding
scalar Laplacian is

$\displaystyle\nabla_{\\!\epsilon_{g}}^{2}$
$\displaystyle=(\epsilon_{g}g_{\epsilon_{g}})^{-1/2}{\partial}_{\mu}\big{(}(\epsilon_{g}g_{\epsilon_{g}})^{1/2}g_{\epsilon_{g}}^{\mu\nu}{\partial}_{\nu}\big{)}$
$\displaystyle=\epsilon_{g}{\rm g}^{-1/2}N^{-1}e_{0}\big{(}{\rm
g}^{1/2}N^{-1}e_{0}\big{)}+{\rm g}^{-1/2}N^{-1}{\partial}_{a}\big{(}N{\rm
g}^{1/2}{\rm
g}^{ab}{\partial}_{b}\big{)}=:\epsilon_{g}\nabla^{2}_{t}+\nabla^{2}_{s}\,.$
(2.39)

Here $e_{0}={\partial}_{t}-{\cal L}_{\vec{N}}$ is the Lie derivative
transversal to the leaves of the foliation. Note that the rightmost $e_{0}$
acts on spatial scalars as $e_{0}(f)={\partial}_{t}f-N^{a}{\partial}_{a}f$,
while the next $e_{0}$ acts on a $+1$ spatial density according to
$e_{0}(\sqrt{{\rm g}}f)={\partial}_{t}(\sqrt{{\rm
g}}f)-{\partial}_{a}(N^{a}\sqrt{{\rm g}}f)$. In $1+d$ form the diffeomorphism
group acts nonlinearly according to the transformation formulas in (2.8) but
for fixed signature parameter $\epsilon_{g}$, – the same in (2.8) and (2.2)–,
$\nabla^{2}_{\\!\epsilon_{g}}$ will continue to map scalars to scalars. The
temporal and spatial parts individually are of course only invariant under
foliation preserving diffeomorphisms. The structure (2.2) carries over to non-
fidual foliations on account of Proposition 2.3.

######  Corollary 2.4.

The complexified Hessian (2.2) is invariant under the complex transformations
(2.12), i.e. $\Delta_{\theta}^{\prime}=\Delta_{\theta}$, in the respective
local coordinates. Also in generic non-fiducial foliations it decomposes
according to $-i\Delta^{\prime}_{\theta}=-\cos\theta\,{\cal
D}^{\prime}_{-}+i\sin\theta\,{\cal D}^{\prime}_{+}$, where ${\cal
D}^{\prime}_{\pm}$ refer to $(N^{\prime},{N^{\prime}}^{a},{\rm
g}^{\prime}_{ab})_{\pm}$ and are separately invariant, ${\cal
D}^{\prime}_{+}={\cal D}_{+}$, ${\cal D}^{\prime}_{-}={\cal D}_{-}$, with
respect to ${\rm transf}_{+}$, ${\rm transf}_{-}$ in (2.8).

Remarks.

(i) We write $-i\Delta_{\theta}$ for the complexified Hessian as
$\Delta_{\theta}$ will generate a semigroup and in the mathematical literature
semigroups arise schematically as $\zeta\mapsto e^{\zeta A}$ (without
prefactors in the exponential) from their generator $A$.

(ii) The background $\varphi$ is not assumed to be on-shell,
$\Delta_{\theta}\varphi\neq 0$. It only indirectly enters the potential $V$,
which is below directly assumed to be a non-negative $C^{\infty}(M)$ function.
Imposing $\Delta_{\theta}\varphi=0$ for any fixed $\theta$ is unproblematic;
its extension to all $\theta$ requires ${\cal D}_{+}\varphi=0={\cal
D}_{-}\varphi$ and thus would allow only simple (e.g. static) backgrounds.
While on-shell backgrounds are commonly used for simplicity, they are not
mandatory in the background field formalism of functional integrals. In
particular, the Legendre effective action $\Gamma[\langle f\rangle,\varphi]$
can consistently be defined for off-shell backgrounds.

In summary, for field theories on real foliated manifolds satisfying the WEC
the above notion of a Wick rotation is fairly satisfactory, if reliant on a
fiducial foliation for its definition. For definiteness we considered only
minimally coupled scalar matter, but the admissibility carries over to many
other standard matter systems like Maxwell fields or perfect fluids. We are
not aware of a manifestly foliation-independent Wick rotation for conventional
field theories on real foliated manifolds.

## 3\. Wick rotated heat semigroup from complexified Hessian

The goal of this section is to establish $\Delta_{\theta}$ as the generator of
an analytic semigroup via suitable resolvent estimates. In the introductory
tabulation this corresponds to the results (a)–(d). Some background material
is prepared in Section 3.1. Before turning to the proofs we add some comments
on the relation to other approaches pursued in the literature to overcome the
limitation of the heat kernel proper to a strictly Euclidean setting.

Often a formal pseudo-heat kernel is introduced by replacing $s>0$ by
$i\tilde{s}$, $\tilde{s}\in\mathbb{R}$, in the diffusion equation [39, 12].
This renders the formal exponential of an (indefinite, unbounded) Lorentzian
signature Hessian ${\cal D}_{-}$ more palatable, but does little in ensuring
the existence of the operator semigroup $\mathbb{R}\ni\tilde{s}\mapsto
e^{-i\tilde{s}{\cal D}_{-}}$. An $i\epsilon$ prescription,
$\tilde{s}\mapsto\tilde{s}-i\epsilon$, alone does not help, unless the
generator is again positive. For a non-elliptic ${\cal D}_{-}$ the key step is
to establish it’s essential self-adjointness under suitable conditions [49,
36, 37], after which the existence of a unique associated Schrödinger group is
ensured.

For a wave operator ${\cal D}_{-}$, in the physics literature only a formal
series realization of the putative Schrödinger kernel of $e^{-i\tilde{s}{\cal
D}_{-}}$, $\tilde{s}\in\mathbb{R}$, is aimed at and invokes an ansatz in terms
of the Synge function and its derivatives. The Synge function is locally well-
defined on a Lorentzian manifold and the recursion relations for the
coefficients in the ansatz are virtually the same as in the Riemannian case
[12]. The solution formulas likewise have no explicit dependence on the
signature parameter. These off-diagonal pseudo-heat kernel coefficients are
moreover in one-to-one correspondence to those occurring in the (Lorentzian
signature) Hadamard expansion [12], so there is little doubt that these
coefficients correctly reflect the short distance behavior of the Hadamard
parametrix. While the Hadamard parametrix can be defined independently of
these series expansions, the use of the inverse Laplace transform to define
the corresponding part of a pseudo-heat kernel is not immediate. The latter
would require information about the resolvent set of ${\cal D}_{-}$, and for a
Lorentzian signature Hessian essential self-adjointness is secondary [3], and
may not always hold [27].

A rigorous result on a near Lorentzian Hadamard expansion is [17], where the
$i\epsilon$ part of the Hadamard parametrix is kept finite and local. Starting
from the original, $i\epsilon$-independent wave equation the regularization
terms are shown to be governed by recursive relations analogous to the
standard ones and the existence of a deformed parametrix is shown. The
relation to a Euclidean regime is not discussed.

One way to link the Euclidean heat kernel to the Lorentzian signature Hadamard
parametrix is by subsuming both in a setting that invokes complex analytic
metrics [34]. This allows for a local Wick rotation (in time), which is
however potentially coordinate dependent. This framework can be used to prove
the symmetry of the coefficients to all orders by transitioning between
different real sections. It also leads to a natural notion of analytic
Hadamard states [20] that extends beyond the stationary case [48]. On the
other hand, a Wick rotation in time may be limited to purely electric metrics
[25].

### 3.1 Background: analytic semigroups

For readability’s sake we include a brief summary of the definitions and
results needed, see e.g. [15] for a detailed account.

######  Definition 3.1 (Bounded analytic semigroup).

Let ${\mathcal{H}}$ be a complex Hilbert space. Given $\alpha\in(0,\pi]$, the
“sector” $\Sigma_{\alpha}\subseteq{\mathbb{C}}$ is defined by222“Arg” is
defined in $(-\pi,\pi)$.

$\displaystyle\Sigma_{\alpha}:=\big{\\{}\zeta\in{\mathbb{C}}\setminus\\{0\\}\,\big{|}\,|{\rm
Arg}(\zeta)|<\alpha\big{\\}}\,.$ (3.1)

Then, a family of bounded operators
$(T(\zeta))_{\zeta\in\Sigma_{\delta}\cup\\{0\\}}$ in
${\mathfrak{B}}({\mathcal{H}})$ (i.e. the space of bounded endomorphisms of
${\mathcal{H}}$) is called a bounded analytic semigroup of angle
$\delta\in(0,\pi/2]$ if:

  1. (i)

$T(0)=\mbox{11}$ and $T(\zeta_{1}+\zeta_{2})=T(\zeta_{1})T(\zeta_{2})$ for all
$\zeta_{1},\,\zeta_{2}\in\Sigma_{\delta}$.

  2. (ii)

The mapping $\zeta\mapsto T(\zeta)$ is analytic in $\Sigma_{\delta}$, i.e.
there is a powerseries expansion about every $\zeta\in\Sigma_{\delta}$ that
converges in operator norm with nonzero radius of convergence.

  3. (iii)

For all $\psi\in{\mathcal{H}}$ and $0<\delta^{\prime}<\delta$ we have strong
continuity, i.e. $\lim_{\Sigma_{\delta^{\prime}}\ni\zeta\to
0}T(\zeta)\psi=\psi$.

  4. (iv)

For every $0<\delta^{\prime}<\delta$, $\norm{T(\zeta)}_{\rm op}$ is uniformly
bounded in $\Sigma_{\delta^{\prime}}$.

Remarks.

(i) Restricting the domain of the bounded analytic semigroup to
${\mathbb{R}}_{\geq 0}\subseteq\Sigma_{\delta}\cup\\{0\\}$ yields a strongly
continuous semigroup $(T(s))_{s\geq 0}$.

(ii) A closed densely defined operator $A:D(A)\to{\mathcal{H}}$,
$D(A)\subseteq{\mathcal{H}}$, is the generator of the analytic semigroup
$(T(\zeta))_{\zeta\in\Sigma_{\delta}\cup\\{0\\}}$ iff

$\displaystyle\forall\,\psi\in D(A):\quad\lim_{s\to
0^{+}}\frac{1}{s}(T(s)\psi-\psi)=A\psi\,,$ (3.2)

i.e. the difference quotient converges in norm (as $\mathbb{R}_{+}\ni s\to
0^{+}$) to $A\psi$ for all $\psi\in D(A)$. The defining property of an
analytic semigroup’s generator is its sectoriality.

######  Definition 3.2.

A densely defined closed operator $A:D(A)\to{\mathcal{H}}$ is called sectorial
of angle $\delta\in(0,\pi/2]$ if:

  1. (i)

$\Sigma_{\pi/2+\delta}\subseteq\rho(A)$, where
$\rho(A):=\big{\\{}\lambda\in{\mathbb{C}}\,\big{|}\,\lambda-A\text{ is
bijective and }(\lambda-A)^{-1}\text{ is bounded}\,\big{\\}}$ is the resolvent
set of $A$.

  2. (ii)

The resolvents are uniformly bounded over sectors, i.e. for every
$\varepsilon\in(0,\delta)$ there exists $M_{\varepsilon}\geq 1$ such that

$\displaystyle\norm{(\lambda-A)^{-1}}_{\rm
op}\leq\frac{M_{\varepsilon}}{|\lambda|}\,,\quad\lambda\in\overline{\Sigma_{\pi/2+\delta-\varepsilon}}\setminus\\{0\\}\,.$
(3.3)

The exponentiation of a sectorial operator (which need not be normal, c.f. the
operator $\Delta_{\theta}$ in (3.10) for generic $\theta$) may be defined via
the holomorphic functional calculus.

######  Theorem 3.1.

Let $(A,D(A))$ be a sectorial operator of angle $\delta\in(0,\pi/2]$ in a
Hilbert space ${\mathcal{H}}$. Consider the following family of operators in
${\mathfrak{B}}({\mathcal{H}})$,

$\displaystyle T(0):=\mbox{11},\quad T(\zeta):=\frac{1}{2\pi
i}\int_{\gamma}\\!e^{\zeta\lambda}(\lambda-A)^{-1}d\lambda\,,\quad\zeta\in\Sigma_{\delta}\,,$
(3.4)

where, given $\zeta\in\Sigma_{\delta}$, $\gamma$ is any piecewise smooth curve
in $\Sigma_{\pi/2+\delta}$ going from $\infty e^{-i(\pi/2+\delta^{\prime})}$
to $\infty e^{i(\pi/2+\delta^{\prime})}$ for some $|{\rm
Arg}(\zeta)|<\delta^{\prime}<\delta$.

Then the family $(T(\zeta))_{\zeta\in\Sigma_{\delta}\cup\\{0\\}}$ is a bounded
analytic semigroup with generator $(A,D(A))$, and is furthermore uniquely
determined the generator.

If, in addition, there is $\varphi\in(0,\delta)$ such that there is the
uniform resolvent bound

$\displaystyle\forall\,\lambda\in\Sigma_{\varphi}:\quad\norm{(\lambda-A)^{-1}}_{\rm
op}\leq\frac{1}{|\lambda|}\,,$ (3.5)

then $T(\zeta)$ is contractive in $\Sigma_{\varphi}\cup\\{0\\}$, i.e.
$\forall\,\zeta\in\Sigma_{\varphi}\cup\\{0\\}:\,\norm{T(\zeta)}_{\rm op}\leq
1$.

The existence of the analytic semigroup and the contractivity result is a
straightforward combination of the Lumer-Phillips Theorem and the generation
theorem for analytic semigroups (resp. Thms. 3.15 and 4.6 in Ch. II of [15]),
while the uniqueness follows from the analyticity of the mapping
$\Sigma_{\delta}\ni\zeta\to T(z)\in{\mathfrak{B}}({\mathcal{H}})$, together
with the fact that the strongly continuous semigroup $(T(s))_{s\geq 0}$
obtained by restricting to $[0,\infty)$ is uniquely determined by its
generator $(A,D(A))$ (c.f. Thm. 1.4 in Ch. II of [15]). We omit the proof, and
instead note the following.

Remarks.

(i) The use of the Cauchy integral formula (3.4), and the fact the semigroup
is uniquely determined by its generator, motivates the intuitive notation
$e^{\zeta A}\equiv T(\zeta)$. For brevity we shall refer to the semigroup
$T(\zeta)$ as the analytic semigroup generated by $A$ and often write
$e^{\zeta A}$ for it.

(ii) The map $\Sigma_{\delta}\ni\zeta\mapsto T(\zeta)\in{\mathfrak{B}}(H)$ is
analytic, and the integral formula (3.4), together with the closedness of
$(A,D(A))$, implies for all $\zeta\in\Sigma_{\delta}$

$\displaystyle\frac{d^{n}}{d\zeta^{n}}T(\zeta)\psi=A^{n}T(\zeta)\psi\,,\quad
n\in{\mathbb{N}}_{0},\,\psi\in{\mathcal{H}}\,.$ (3.6)

In particular, this entails boundedness of the $A^{n}T(\zeta)$, and that
$\text{ran}(T(\zeta))\subseteq\cap_{n=1}^{\infty}D(A^{n})$.

(iii) Moreover, a straightforward adaptation of the arguments in [15] shows
that for all $\delta^{\prime}\in(0,\delta)$ there is a $C_{\delta^{\prime}}>0$
such that

$\displaystyle\forall\,\zeta\in\Sigma_{\delta^{\prime}}:\quad\norm{AT(\zeta)}_{\text{op}}\leq\frac{C_{\delta^{\prime}}}{|\zeta|}\,,$
(3.7)

Observing that $A^{n}T(\zeta)=(AT(\zeta/n))^{n}$, it follows that for all
$n\in{\mathbb{N}}$,

$\displaystyle\forall\,\zeta\in\Sigma_{\delta^{\prime}}:\quad\norm{A^{n}T(\zeta)}_{\text{op}}\leq\Big{(}\frac{nC_{\delta^{\prime}}}{|\zeta|}\Big{)}^{n}\,,$
(3.8)

which will be needed later on. Finally, for any $\zeta_{0}\in\Sigma_{\delta}$,
the (convergent) Taylor expansion
$T(\zeta)=\sum_{k=0}^{\infty}\frac{(\zeta-\zeta_{0})^{k}}{k!}T^{(k)}(\zeta_{0})$,
together with (3.6) implies that for any $\psi\in{\mathcal{H}}$

$\displaystyle
A^{n}T(\zeta)\psi=\sum_{k=0}^{\infty}\frac{(\zeta-\zeta_{0})^{k}}{k!}A^{n+k}T(\zeta_{0})\psi\,,\quad
n\in{\mathbb{N}}_{0}\,,$ (3.9)

with the sum converging in norm within the disk of convergence.

### 3.2 Spectrum and sectoriality of the complexified Hessian

Returning now to the complexified Hessian $-i\Delta_{\theta}$ in (2.2), we
show that $\Delta_{\theta}$ (defined on an appropriate dense domain of
$L^{2}(M)$) is a sectorial operator, and hence generates an analytic semigroup
for every $\theta\in(0,\pi)$. In more detail, throughout this section, we
focus on the family of operators

$\displaystyle\Delta_{\theta}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\sin\theta\,{\cal D}_{+}-i\cos\theta\,{\cal
D}_{-}\,,\quad\theta\in(0,\pi)\,,$ (3.10)

where ${\mathcal{D}}_{\pm}:=-\nabla_{\pm}^{2}+V$ are the Euclidean/Lorentzian
signature Hessians and $V\in C^{\infty}(M)$ is non-negative and bounded (in
particular, ${\mathcal{D}}_{+}$ defined on $C_{c}^{\infty}(M)$ is a positive
operator). Concerning the dense domain $D(\Delta_{\theta})$ in the Hilbert
space $L^{2}(M)$, it is tempting to work with smooth functions and define the
(classical) action of $\Delta_{\theta}$ on $C_{c}^{\infty}(M)$, which is
densely contained in $L^{2}(M)$. However, this domain is unsuitable for
sectoriality as even $\mbox{11}-\Delta_{\theta}$ fails to be surjective.
Instead, we sacrifice classical differentiability and define the domain
$D(\Delta_{\theta})$ to be the following subset of the Sobolev space
${\mathcal{W}}_{0}^{1}(M)$ (see Appendix B)

$\displaystyle\forall\,\theta\in(0,\pi):\,\,D(\Delta_{\theta}):=\big{\\{}u\in{\mathcal{W}}^{1}_{0}(M)\,\big{|}\,\Delta_{\theta}u\in
L^{2}(M)\big{\\}}\,.$ (3.11)

Remarks.

(i) First note that $u\in{\mathcal{W}}^{1}_{0}(M)\overset{\rm
dense}{\hookrightarrow}L^{2}(M)$ defines a distribution
$\widetilde{u}\in{\mathfrak{D}}^{\prime}(M)$ on the space of test functions
${\mathfrak{D}}(M)\equiv C_{c}^{\infty}(M)$, and further the distributional
action of the operator $\Delta_{\theta}$ determines a distribution
$\widetilde{\Delta_{\theta}u}\in{\mathfrak{D}}^{\prime}(M)$, c.f. Appendix B.
Then the condition $\Delta_{\theta}u\in L^{2}(M)$ in (3.11) means that there
exists $h\in L^{2}(M)$ such that

$\displaystyle\forall\,w\in
C_{c}^{\infty}(M):\,\,\int\\!d\mu_{g}\,u\Delta_{\theta}w=\int\\!\,d\mu_{g}\,hw\,,$
(3.12)

with $\mu_{g}$ the measure on $M$ defined by the real metric volume element
$N\sqrt{{\rm g}}$, c.f. Section 2.2; once (3.12) holds, one defines
$\Delta_{\theta}u:=h$.

(ii) Clearly $C_{c}^{\infty}(M)\subseteq D(\Delta_{\theta})\subseteq
L^{2}(M)$, so $D(\Delta_{\theta})$ is dense in $L^{2}(M)$.

(iii) Finally, although $D(\Delta_{\theta})$ contains non-classically
differentiable functions, one may still integrate by parts: for all $u,v\in
D(\Delta_{\theta})$

$\displaystyle\int\\!d\mu_{g}\,u(\sin\theta\nabla_{+}^{2}+i\cos\theta\nabla_{-}^{2})v=-\int\\!d\mu_{g}\,(\sin\theta
g^{\alpha\beta}_{+}+i\cos\theta
g^{\alpha\beta}_{-}){\partial}_{\alpha}u{\partial}_{\beta}v\,.$ (3.13)

This follows readily from approximation with $C_{c}^{\infty}(M)$ functions in
${\mathcal{W}}^{1}$-norm, together with the definitions of the distributional
gradient and the distributional action of
$\sin\theta\nabla_{+}^{2}+i\cos\theta\nabla_{-}^{2}$; we omit a detailed
proof.

We now state and prove the main result of this section.

######  Theorem 3.2.

Let $(M,g_{-})$ be a globally hyperbolic manifold, $\theta\in(0,\pi)$, and
$\tilde{\theta}:=\min\\{\theta,\pi-\theta\\}$. Selecting a fiducial foliation
on $(M,g_{-})$ with ADM triple $(N,N^{a},{\rm g}_{ab})_{-}$ and performing a
lapse Wick rotation $(N,N^{a},{\rm
g}_{ab})_{-}\mapsto(e^{-i\theta}N,N^{a},{\rm g}_{ab})_{-}$ to define a
complexified metric tensor, consider the operator
$(\Delta_{\theta},D(\Delta_{\theta}))$ in $L^{2}(M)$ defined by

$\displaystyle\Delta_{\theta}:=-\sin\theta{\mathcal{D}}_{+}-i\cos\theta{\mathcal{D}}_{-}\,,\quad{\mathcal{D}}_{\pm}:=-\nabla_{\pm}^{2}+V\,,$
(3.14)

with $V\in C^{\infty}(M)$ non-negative and bounded, and domain determined by
its distributional action,

$\displaystyle
D(\Delta_{\theta}):=\\{u\in{\mathcal{W}}_{0}^{1}(M)\;|\;\Delta_{\theta}u\in
L^{2}(M)\\}\,.$ (3.15)

Then $(\Delta_{\theta},D(\Delta_{\theta}))$ generates a unique bounded
analytic semigroup
$(e^{\zeta\Delta_{\theta}})_{\zeta\in\Sigma_{\tilde{\theta}}\cup\\{0\\}}$ that
is contractive, i.e. $||e^{\zeta\Delta_{\theta}}||_{\text{op}}\leq 1$ for all
$\zeta\in\Sigma_{\tilde{\theta}}\cup\\{0\\}$.

Further, the adjoint semigroup is given by

$\displaystyle\forall\,\zeta\in\Sigma_{\tilde{\theta}}:\,\,(e^{\zeta\Delta_{\theta}})^{\ast}=e^{\zeta^{\ast}\Delta_{\pi-\theta}}\,.$
(3.16)

We prepare the following lemma, whose proof is postponed to the end of the
section.

######  Lemma 3.3.

For $\theta\in(0,\pi)$, and $\tilde{\theta}:=\min\\{\theta,\pi-\theta\\}$, the
resolvent set $\rho(\Delta_{\theta})$ contains the sector
$\Sigma_{\pi/2+\tilde{\theta}}$, i.e.
$\Sigma_{\pi/2+\tilde{\theta}}\subseteq\rho(\Delta_{\theta})$. Equivalently,
the spectrum $\sigma(\Delta_{\theta})$ is contained in the (closed) set
${\mathbb{C}}\setminus\Sigma_{\pi/2+\tilde{\theta}}$. Moreover, for each
$\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$ there is a constant
$C_{\tilde{\theta}^{\prime}}\geq 1$ such that

$\displaystyle\forall\,\lambda\in\Sigma_{\pi/2+\tilde{\theta}^{\prime}}:\quad\norm{(\lambda-\Delta_{\theta})^{-1}}_{\rm
op}\leq\frac{C_{\tilde{\theta}^{\prime}}}{|\lambda|}\,,$ (3.17)

and further

$\displaystyle\forall\,\lambda\in\Sigma_{\tilde{\theta}}:\quad\norm{(\lambda-\Delta_{\theta})^{-1}}_{\rm
op}\leq\frac{1}{|\lambda|}\,.$ (3.18)

Proof of Theorem 3.2. Lemma 3.3 entails that
$(\Delta_{\theta},D(\Delta_{\theta}))$ is sectorial of angle $\tilde{\theta}$,
and accordingly generates a unique bounded analytic semigroup
$(e^{\zeta\Delta_{\theta}})_{\zeta\in\Sigma_{\tilde{\theta}}\cup\\{0\\}}$ by
Theorem 3.1. Moreover, the resolvent bound (3.18) of Lemma 3.3 for all
$\lambda\in\Sigma_{\tilde{\theta}}$ implies
$(e^{\zeta\Delta_{\theta}})_{\zeta\in\Sigma_{\tilde{\theta}}\cup\\{0\\}}$ is a
family of contractive operators.

Next, the adjoint result (3.16) is an immediate corollary of the fact

$\displaystyle\Delta_{\theta}^{\ast}=\Delta_{\pi-\theta}\,,\quad
D(\Delta_{\theta}^{\ast})=D(\Delta_{\pi-\theta})\,,$ (3.19)

which we now prove. Since the all elements of $D(\Delta_{\theta})$ and
$D(\Delta_{\pi-\theta})$ are Sobolev ${\mathcal{W}}_{0}^{1}$-functions,
integration by parts (c.f. (3.13)) gives
$\Delta_{\pi-\theta}\subseteq\Delta_{\theta}^{\ast}$ (i.e.
$D(\Delta_{\pi-\theta})\subseteq D(\Delta_{\theta}^{\ast})$ and
$\Delta_{\theta}^{\ast}|_{D(\Delta_{\pi-\theta})}=\Delta_{\pi-\theta})$. In
order to show equality, let $\lambda>0$ and define the operator
$B_{\theta}:=\lambda-\Delta_{\theta}$ with domain
$D(B_{\theta}):=D(\Delta_{\theta})$. By definition of the adjoint, $f\in
D(B^{\ast}_{\theta})$ iff there exists a unique $g\in L^{2}(M)$ such that for
all $h\in D(B_{\theta})$ we have $\left\langle
f\,\middle|\,B_{\theta}h\right\rangle_{L^{2}(M)}=\left\langle
g\,\middle|\,h\right\rangle_{L^{2}(M)}$, and by definition
$B_{\theta}^{\ast}f:=g$. On the other hand, Lemma 3.3 implies that
$B_{\pi-\theta}^{-1}=(\lambda-\Delta_{\pi-\theta})^{-1}\in{\mathfrak{B}}(L^{2}(M))$,
so there is $f^{\prime}:=(\Delta_{\pi-\theta}-\lambda)^{-1}g\in
D(\Delta_{\pi-\theta})$. Hence, for any $h\in D(B_{\theta})$

$\displaystyle\left\langle
B_{\theta}^{\ast}f\,\middle|\,h\right\rangle_{L^{2}(M)}=\left\langle(\Delta_{\pi-\theta}-\lambda)f^{\prime}\,\middle|\,h\right\rangle_{L^{2}(M)}=\left\langle
f^{\prime}\,\middle|\,(\Delta_{\theta}-\lambda)h\right\rangle_{L^{2}(M)}\,,$
(3.20)

where the second equality arises from integrating by parts as
$f^{\prime},\,h\in{\mathcal{W}}^{1}_{0}(M)$ (c.f. (3.13)). Comparing $f$ and
$f^{\prime}$, it follows that for all $h\in D(B_{\theta})$ we have the result
$\left\langle
f-f^{\prime}\,\middle|\,(\Delta_{\theta}-\lambda)h\right\rangle_{L^{2}(M)}=0$,
or equivalently
$f-f^{\prime}\in\text{ran}(\lambda-\Delta_{\theta})^{\perp}=L^{2}(M)^{\perp}$,
and hence $f=f^{\prime}\in D(\Delta_{\pi-\theta})$. So
$B_{\theta}^{\ast}=(\lambda-\Delta_{\theta})^{\ast}=\lambda-\Delta_{\pi-\theta}$
holds for all $\lambda>0$, which implies
$\Delta_{\theta}^{\ast}=\Delta_{\pi-\theta}$. ∎

Proof of Lemma 3.3. Fixing an arbitrary $\theta\in(0,\pi)$, we begin by
establishing $\Sigma_{\pi/2+\tilde{\theta}}\subseteq\rho(\Delta_{\theta})$.
Namely, given $\lambda\in\Sigma_{\pi/2+\tilde{\theta}}$, it is to be shown
that $\lambda-\Delta_{\theta}:D(\Delta_{\theta})\to L^{2}(M)$ has a bounded
inverse.

Surjectivity: This follows through an application of the Lax-Milgram lemma
(Ch. 6 Theorem 6 in [30]). Fixing $f\in L^{2}(M)$, one defines the antilinear
bounded map $\Psi_{f}:{\mathcal{W}}_{0}^{1}(M)\to{\mathbb{C}}$ by

$\Psi_{f}(w):=\int\\!d\mu_{g}\,fw^{\ast}\,,$ (3.21)

and the sesquilinear form
${\mathcal{E}}:{\mathcal{W}}_{0}^{1}(M)\times{\mathcal{W}}_{0}^{1}(M)\to{\mathbb{C}}$
mapping $u,w\in{\mathcal{W}}^{1}_{0}(M)$ to

$\displaystyle{\mathcal{E}}(u,w):=\int\\!d\mu_{g}\,\Big{[}(\lambda+ie^{-i\theta}V)uw^{\ast}+(\sin\theta
g_{+}^{\alpha\beta}+i\cos\theta
g_{-}^{\alpha\beta}){\partial}_{\alpha}u{\partial}_{\beta}w^{\ast}\Big{]}\in{\mathbb{C}}\,.$
(3.22)

Since the potential function $V$ is bounded on $M$, an application of the
Cauchy-Schwarz inequality and the identity333This follows by direct
computation from $ds^{2}_{\epsilon_{g}}=\epsilon_{g}N^{2}dt^{2}+{\rm
g}_{ab}(dx^{a}+N^{a}dt)(dx^{b}+N^{b}dt)$.
$g_{-}^{\mu\sigma}g_{\sigma\rho}^{+}g_{-}^{\rho\nu}=g_{+}^{\mu\nu}$ shows that
${\mathcal{E}}$ is bounded, i.e.,
$\exists\,c>0:\,\forall\,u,w\in{\mathcal{W}}_{0}^{1}(M):\,|{\mathcal{E}}(u,w)|\leq
c\norm{u}_{{\mathcal{W}}^{1}(M)}\norm{w}_{{\mathcal{W}}^{1}(M)}$. Then, if
${\mathcal{E}}$ is coercive, i.e.,
$\exists\,\alpha>0:\,\forall\,u\in{\mathcal{W}}^{1}_{0}(M):\,|{\mathcal{E}}(u,u)|\geq\alpha\norm{u}_{{\mathcal{W}}^{1}}$,
the Lax-Milgram lemma implies the existence of a unique
$v\in{\mathcal{W}}_{0}^{1}(M)$ such that

$\displaystyle\forall\,w\in{\mathcal{W}}^{1}_{0}(M):\,\,{\mathcal{E}}(v,w)=\Psi_{f}(w)\,.$
(3.23)

To establish the coercivity bound, consider the restriction of ${\mathcal{E}}$
to the unit sphere
${\mathcal{S}}_{{\mathcal{W}}_{0}^{1}}:=\\{w\in{\mathcal{W}}_{0}^{1}(M)\;|\;\norm{w}_{{\mathcal{W}}^{1}(M)}=1\\}$.
Then, we may express

$\displaystyle{\mathcal{E}}(w,w)=(1-e(w))\lambda-\Big{[}-\sin\theta\,e(w)-i\cos\theta\,\ell(w)-ie^{-i\theta}\int\\!d\mu_{g}\,V|w|^{2}\Big{]}\,,$
(3.24)

with

$\displaystyle
e(w):=\int\\!d\mu_{g}\,g_{+}^{\alpha\beta}{\partial}_{\alpha}w{\partial}_{\beta}w^{\ast}\,,\quad\ell(w):=\int\\!d\mu_{g}\,g_{-}^{\alpha\beta}{\partial}_{\alpha}w{\partial}_{\beta}w^{\ast}\,.$
(3.25)

Clearly, $e(w)\geq 0$, $\ell(w)\in{\mathbb{R}}$ and the bound $|\ell(w)|\leq
e(w)$ is easily seen; hence the two terms on the right hand side of (3.24) are
elements of disjoint sets,

$\displaystyle(1-e(w))\lambda\in\Sigma_{\pi/2+\tilde{\theta}}\,,$
$\displaystyle-\sin\theta\,e(w)-i\cos\theta\,\ell(w)-ie^{-i\theta}\int\\!d\mu_{g}\,V|w|^{2}\in{\mathbb{C}}\setminus\Sigma_{\pi/2+\tilde{\theta}}\,.$
(3.26)

It then follows that these terms are bounded away from each other uniformly
over ${\mathcal{S}}_{{\mathcal{W}}_{0}^{1}}$, and so there exists $\alpha>0$
such that for all
$w\in{\mathcal{S}}_{{\mathcal{W}}_{0}^{1}}:\,|E(w,w)|\geq\alpha$. Rescaling
yields the requisite coercivity bound.

Having established coercivity, $\Psi_{f}$ is uniquely realized by
${\mathcal{E}}$ through the relation (3.23). In particular, for any $w\in
C_{c}^{\infty}(M)\subseteq{\mathcal{W}}_{0}^{1}(M)$, integrating (3.23) by
parts yields

$\displaystyle\int\\!d\mu_{g}\,fw^{\ast}=\int\\!d\mu_{g}\,v(\lambda-\Delta_{\theta})w^{\ast}\,.$
(3.27)

In summary, given $f\in L^{2}(M)$, there exists a unique
$v\in{\mathcal{W}}_{0}^{1}(M)$ such that $\Delta_{\theta}v=\lambda v-f\in
L^{2}(M)$ and $(\lambda-\Delta_{\theta})v=f$, establishing surjectivity.

Injectivity: For any element $w\in{\mathcal{S}}^{\prime}:=\\{u\in
D(\Delta_{\theta})\;|\;\norm{u}_{L^{2}}=1\\}$,

$\displaystyle\norm{(\lambda-\Delta_{\theta})w}_{L^{2}(M)}$
$\displaystyle\geq\big{|}\left\langle
w\,\middle|\,(\lambda-\Delta_{\theta})w\right\rangle_{L^{2}(M)}\big{|}\,,$
$\displaystyle=\lambda-\Big{[}-\sin\theta
e(w)-i\cos\theta\ell(w)-ie^{-i\theta}\int\\!d\mu_{g}\,V|w|^{2}\Big{]}\,,$
(3.28)

where the second line follows from an integration by parts, and $e(w)$,
$\ell(w)$ are defined in (3.25). This is the difference of two terms,
$\lambda\in\Sigma_{\pi/2+\tilde{\theta}}$, and the other an element of
${\mathbb{C}}\setminus\Sigma_{\pi/2+\tilde{\theta}}$. Then, clearly for all
$w\in{\mathcal{S}}^{\prime}$

$\displaystyle\big{|}\langle
w|(\lambda-\Delta_{\theta})w\rangle\big{|}\geq\text{dist}(\lambda,\,{\mathbb{C}}\setminus\Sigma_{\pi/2+\theta})=:d_{\lambda}(\theta)>0\,,$
(3.29)

implying injectivity.

Resolvent bounds: For $\lambda\in\Sigma_{\pi/2+\tilde{\theta}}$, the bijection
$\lambda-\Delta_{\theta}:D(\Delta_{\theta})\to L^{2}(M)$ has inverse
$(\lambda-\Delta_{\theta})^{-1}$, and (3.29) implies the resolvent bound

$\displaystyle\norm{(\lambda-\Delta_{\theta})^{-1}}_{\rm
op}\leq\frac{1}{d_{\lambda}(\theta)}\,.$ (3.30)

When $\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$, for any
$\lambda\in\Sigma_{\pi/2+\theta^{\prime}}$ one has
$d_{\lambda}(\theta)\geq{\rm
sin}(\tilde{\theta}-\tilde{\theta}^{\prime})|\lambda|$, yielding the uniform
bound

$\displaystyle\norm{(\lambda-\Delta_{\theta})^{-1}}_{\rm op}\leq\frac{1}{{\rm
sin}(\tilde{\theta}-\tilde{\theta}^{\prime})}\,\frac{1}{|\lambda|}\,,\quad\forall\,\lambda\in\Sigma_{\pi/2+\tilde{\theta}^{\prime}}\,,$
(3.31)

Setting $C_{\tilde{\theta}^{\prime}}:={\rm
sin}(\tilde{\theta}-\tilde{\theta}^{\prime})^{-1}\geq 1$ gives the result
(3.17). On the other hand, restricting to
$\lambda\in\Sigma_{\tilde{\theta}^{\prime}}$, it is easy to see that
$d_{\lambda}(\theta)=|\lambda|$. Thus (3.30) yields the result (3.18),
completing the proof of Lemma 3.3. ∎

## 4\. Semigroup action and the Wick rotated heat kernel

Two of the key features of the standard heat semigroup associated to the
Dirichlet Laplacian on Riemannian manifolds are its smoothening property and
its realizability as an integral operator with a smooth kernel. Having shown
the existence of the Wick rotated heat semigroup for all $\theta\in(0,\pi)$ in
the previous section, we collect a number of results on the semigroup’s action
in Theorem 4.1, in particular its smoothening behavior. The second main result
of this section, Theorem 4.2, realizes the semigroup in terms of a unique,
smooth integral kernel,

$\displaystyle\big{(}e^{\zeta\Delta_{\theta}}\psi\big{)}(y)=\int\\!d\mu_{g}(y^{\prime})\,K^{\theta}_{\zeta}(y,y^{\prime})\psi(y^{\prime})\,,$
$\displaystyle({\partial}_{\zeta}-\Delta_{\theta,y})K^{\theta}_{\zeta}(y,y^{\prime})=0\,,\quad\lim_{\zeta\rightarrow
0}K_{\zeta}^{\theta}(y,y^{\prime})=\delta(y,y^{\prime})\,.$ (4.1)

Recall that $\mu_{g}$ is the $\theta$-independent measure arising from the
real metric volume form $dyN\\!\sqrt{{\rm g}}$, and the $\delta$-distribution
is normalized with respect to it.

######  Theorem 4.1.

Let $\theta\in(0,\pi)$ and $\tilde{\theta}:=\min\\{\theta,\pi-\theta\\}$.

  1. (i)

For any $\psi\in L^{2}(M)$ consider the map
$\Sigma_{\tilde{\theta}}\ni\zeta\mapsto e^{\zeta\Delta_{\theta}}\psi\in
L^{2}(M)$. Then, for each $\zeta\in\Sigma_{\tilde{\theta}}$, the image
$e^{\zeta\Delta_{\theta}}\psi\in L^{2}(M)$ has a
$C^{\infty}(M)$-representative, denoted by
$(e^{\zeta\Delta_{\theta}}\psi)(\cdot)$, and the mapping

$\displaystyle\Sigma_{\tilde{\theta}}\times
M\ni(\zeta,y)\mapsto(e^{\zeta\Delta_{\theta}}\psi)(y)\in{\mathbb{C}}$ (4.2)

is jointly smooth, and is analytic in $\zeta\in\Sigma_{\tilde{\theta}}$ for
fixed $y\in M$. Moreover, the mapping (4.2) is a classical solution to the
differential equation

$\displaystyle\begin{cases}({\partial}_{\zeta}-\Delta_{\theta})f(\zeta,y)=0\,,\quad\forall\,(\zeta,y)\in\Sigma_{\tilde{\theta}}\times
M\,,\\\\[5.69054pt] f(0,y)=\psi(y)\,,\quad\text{for a.e. }y\in
M\,.\end{cases}$ (4.3)

  2. (ii)

Consider an arbitrary $\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$ and
$\psi\in L^{2}(M)$. Then, for any chart $U$ on $M$, and each open set
$\Omega\Subset U$, there exist for every $m\in{\mathbb{N}}_{0}$ constants
$c_{m}>0$ and $\sigma_{m}\in{\mathbb{N}}$ such that in local coordinates there
are the uniform bounds444The $\norm{\cdot}_{C^{m}(\Omega)}$-norm is defined in
the local chart coordinates by
$\norm{u}_{C^{m}(\Omega)}:=\underset{|\alpha|\leq
m}{\sup}\underset{\Omega_{0}}{\sup}\,|{\partial}^{\alpha}u|$.

$\displaystyle\forall\,\zeta\in\Sigma_{\tilde{\theta}^{\prime}}\,,\,\forall\,m\in{\mathbb{N}}_{0}:\quad\norm{e^{\zeta\Delta_{\theta}}\psi}_{C^{m}(\Omega)}\leq
c_{m}(1+|\zeta|^{-\sigma_{m}})\norm{\psi}_{L^{2}(M)}\,.$ (4.4)

The constant $c_{m}>0$ depends on $\theta^{\prime},\,\theta,\,\Omega,\,U$, but
is independent of $\psi$ and $\zeta$, and $\sigma_{m}\in{\mathbb{N}}$ is
defined as the smallest integer greater than $m/2+(d+1)/4$.

  3. (iii)

If $\psi\in C_{c}^{\infty}(M)$ and
$\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$, then for any chart $U$ on $M$,
and each open set $\Omega\Subset U$, in local coordinates

$\displaystyle\forall\,m\in{\mathbb{N}}_{0}:\quad\norm{e^{\zeta\Delta_{\theta}}\psi-\psi}_{C^{m}(\Omega)}\to
0\,,\quad\text{as}\quad\Sigma_{\tilde{\theta}^{\prime}}\ni\zeta\to 0\,.$ (4.5)

Proof of Theorem 4.1.

(i) Fix an open ball
$B_{S}(\zeta_{0}):=\\{\zeta\in{\mathbb{C}}\;|\;|\zeta-\zeta_{0}|<S\\}\subseteq\Sigma_{\tilde{\theta}}$,
a chart $U$ on $M$, and an open set $\Omega\Subset U$. Integrating in local
coordinates in the chart $U$, we have the bound

$\displaystyle\int_{\Omega}\\!dy\,|\Delta_{\theta}f|^{2}\leq
C^{2}\int_{\Omega}\\!d\mu_{g}|\,\Delta_{\theta}f|^{2}\leq
C^{2}\norm{\Delta_{\theta}f}_{L^{2}(M)}^{2}\,,$ (4.6)

where $C^{2}:=\sup_{\Omega}(N\sqrt{{\rm g}})^{-1/2}<\infty$, since
$\Omega\Subset U$.

Next, given $\psi\in L^{2}(M)$, Eq. (4.6) and the bound (3.8) imply that

$\displaystyle\forall\,n\in{\mathbb{N}}_{0}:\quad\Delta_{\theta}^{n}\,e^{\zeta_{0}\Delta_{\theta}}\psi\in
L^{2}_{\rm loc}(U)\,,$ (4.7)

where $U$ is regarded as an open subset of ${\mathbb{R}}^{d+1}$ via the chart
map, and $L^{2}_{\rm loc}(U)$ is defined relative to the Lebesgue measure $dy$
on ${\mathbb{R}}^{d+1}$. Further, (3.9) entails the Taylor expansion

$\displaystyle\Delta_{\theta}^{n}e^{\zeta\Delta_{\theta}}\psi=\sum_{k=0}^{\infty}\frac{(\zeta-\zeta_{0})^{k}}{k!}\Delta_{\theta}^{n+k}e^{\zeta_{0}\Delta_{\theta}}\psi,$
(4.8)

converging in $L^{2}(M)$ for all $\zeta\in B_{S}(\zeta_{0})$ (with $S>0$
sufficiently small), and hence converging in $L^{2}_{\text{loc}}(U)$ in local
coordinates on the chart $U$. Since in local coordinates, $\Delta_{\theta}$ is
an operator in ${\mathbb{R}}^{d+1}$ of the type (C.1b) considered in Appendix
C, Theorem C.2 implies that for each $\zeta\in B_{S}(\zeta_{0})$, there exists
a $C^{\infty}(U)$ representative of $e^{\zeta\Delta_{\theta}}\psi$ in the
chart, and moreover that the mapping

$\displaystyle B_{S}(\zeta_{0})\times
U\ni(\zeta,y)\mapsto(e^{\zeta\Delta_{\theta}}\psi)(y)\in{\mathbb{C}}$ (4.9)

is jointly smooth. Since both the open ball
$B_{S}(\zeta_{0})\subseteq\Sigma_{\tilde{\theta}}$ and the chart $U$ on $M$
was arbitrary, this establishes that the mapping
$\Sigma_{\tilde{\theta}}\times M\to{\mathbb{C}}$ given by
$(e^{\zeta\Delta_{\theta}}\psi)(y)$ is smooth. Moreover, for each $y\in M$,
the pointwise convergence of (4.8) implies analyticity in
$\zeta\in\Sigma_{\tilde{\theta}}$. Finally, recall that when regarded as an
element of $L^{2}(M)$, $e^{\zeta\Delta_{\theta}}\psi$ satisfies
$({\partial}_{\zeta}-\Delta_{\theta})e^{\zeta\Delta_{\theta}}\psi=0$ and
$\lim_{\Sigma_{\tilde{\theta}}\ni\zeta\to
0}e^{\zeta\Delta_{\theta}}\psi=\psi$. The smoothness of $(\zeta,y)\mapsto
e^{\zeta\Delta_{\theta}}\psi(y)$ then clearly entails it satisfies the
differential equation (4.3).

(ii) Proceeding next to the local coordinate bound, fix arbitrary
$\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$ and open sets
$\Omega\Subset\Omega^{\prime}\Subset U$ (since $\Omega\Subset U$, it is always
possible to find such an open set $\Omega^{\prime}$). Then by Corollary C.1.1
we have for any $\zeta\in\Sigma_{\tilde{\theta}^{\prime}}$ the bound

$\displaystyle\norm{e^{\zeta\Delta_{\theta}}\psi}_{C^{m}(\Omega)}$
$\displaystyle\leq
B\sum_{j=0}^{\sigma_{m}}\norm{\Delta_{\theta}^{j}e^{\zeta\Delta_{\theta}}\psi}_{L^{2}(\Omega^{\prime},\,dy)}\leq
B^{\prime}\sum_{j=0}^{\sigma_{m}}\norm{\Delta_{\theta}^{j}e^{\zeta\Delta_{\theta}}\psi}_{L^{2}(M)}$
$\displaystyle\leq
B^{\prime\prime}\big{(}1+\sum_{j=1}^{\sigma_{m}}(j/|\zeta|)^{j}\big{)}\norm{\psi}_{L^{2}(M)}\leq
c_{m}\big{(}1+|\zeta|^{-\sigma_{m}}\big{)}\norm{\psi}_{L^{2}(M)}$ (4.10)

where (3.8) is used for the penultimate inequality, and
$\sum_{j=1}^{N}(j/|\zeta|)^{j}\leq\sum_{j=1}^{N}j^{j}(1+|\zeta|^{-N})$ for the
final bound.

(iii) Strong continuity of
$(e^{\zeta\Delta_{\theta}})_{\zeta\in\Sigma_{\tilde{\theta}}\cup\\{0\\}}$
(c.f. Definition 3.1(iii)) entails that for any
$\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$ and $\psi\in L^{2}(M)$,
$||e^{\zeta\Delta_{\theta}}\psi-\psi||_{L^{2}(M)}\to 0$ as
$\Sigma_{\tilde{\theta}}\ni\zeta\to 0$. In particular, if $\psi\in
C^{\infty}_{c}(M)$, then for each
$k\in{\mathbb{N}}_{0}:\,\Delta_{\theta}^{k}\psi\in C^{\infty}_{c}(M)$, and
hence $\forall\,k\in{\mathbb{N}}_{0}$

$\displaystyle\norm{\Delta_{\theta}^{k}(e^{\zeta\Delta_{\theta}}\psi-\psi)}_{L^{2}(M)}=\norm{e^{\zeta\Delta_{\theta}}(\Delta_{\theta}^{k}\psi)-\Delta_{\theta}^{k}\psi}_{L^{2}(M)}\to
0\quad\text{as }\Sigma_{\tilde{\theta}}\ni\zeta\to 0\,,$ (4.11)

where the equality follows from the semigroup commuting with its generator.
Corollary C.1.1 then implies the result (4.5). ∎

We proceed to the kernel theorem.

######  Theorem 4.2.

Let $\theta\in(0,\pi)$ and $\tilde{\theta}:=\min\\{\theta,\pi-\theta\\}$. Then
for all $\zeta\in\Sigma_{\tilde{\theta}}$ there is a unique integral kernel
$K_{\zeta}^{\theta}\in C^{\infty}(M\times M)$ such that for all $\psi\in
L^{2}(M)$

$\displaystyle(e^{\zeta\Delta_{\theta}}\psi)(y)=\int\\!d\mu_{g}(y^{\prime})\,K_{\zeta}^{\theta}(y,y^{\prime})\psi(y^{\prime})\,,\quad\forall\,y\in
M\,.$ (4.12)

This integral kernel has the following properties:

  1. (i)

(Hermiticity) For all $(\zeta,y,y^{\prime})\in\Sigma_{\tilde{\theta}}\times
M\times
M:\,K_{\zeta}^{\theta}(y,y^{\prime})=K_{\zeta^{\ast}}^{\pi-\theta}(y^{\prime},y)^{\ast}$.

  2. (ii)

(Regularity) The mapping

$\displaystyle\Sigma_{\tilde{\theta}}\times M\times
M\ni(\zeta,y,y^{\prime})\mapsto
K_{\zeta}^{\theta}(y,y^{\prime})\in{\mathbb{C}}$ (4.13)

is jointly $C^{\infty}$-smooth. Moreover for fixed $(y,y^{\prime})\in M\times
M$, $K_{\zeta}^{\theta}(y,y^{\prime})$ is analytic in
$\zeta\in\Sigma_{\tilde{\theta}}$.

  3. (iii)

(Uniqueness) If for any $\zeta\in\Sigma_{\tilde{\theta}}$ there exists another
integral kernel
$J^{\theta}_{\zeta}\in C^{\infty}(M\times M)$ realizing the semigroup action
via (4.12), then $K_{\zeta}^{\theta}=J^{\theta}_{\zeta}$.

  4. (iv)

(Semigroup property) For all $y,y^{\prime}\in M$ and
$\zeta_{1},\zeta_{2}\in\Sigma_{\tilde{\theta}}$

$\displaystyle
K_{\zeta_{1}+\zeta_{2}}^{\theta}(y,y^{\prime})=\int\\!d\mu_{g}(y^{\prime\prime})\,K_{\zeta_{1}}^{\theta}(y,y^{\prime\prime})K_{\zeta_{2}}^{\theta}(y^{\prime\prime},y^{\prime})\,.$
(4.14)

  5. (v)

(Wick rotated heat equation) For any fixed $y^{\prime}\in M$,

$\displaystyle({\partial}_{\zeta}-\Delta_{\theta,y})K_{\zeta}^{\theta}(y,y^{\prime})=0\quad\text{in
}\Sigma_{\tilde{\theta}}\times M\,.$ (4.15)

Furthermore if $\psi\in C_{c}^{\infty}(M)$ and
$\tilde{\theta}^{\prime}\in(0,\tilde{\theta})$, then for any chart $U$ on $M$,
and each open set $\Omega\Subset U$, in local coordinates we have

$\displaystyle\int\\!d\mu_{g}(y)\,K_{\zeta}^{\theta}(\cdot,y)\psi(y)\longrightarrow\psi(\cdot)$
(4.16)

in $C^{m}(\Omega)$ as $\Sigma_{\tilde{\theta}^{\prime}}\ni\zeta\to 0$.

Remark. These properties mirror closely those of the heat kernel
$K_{s}(y,y^{\prime})$ on a Riemannian manifold $(M,g^{+})$. Additionally, the
latter often satisfies $\int\\!d\mu_{g}(y^{\prime})K_{s}(y^{\prime},y)=1$, for
all $y\in M$, referred to as stochastic completeness of $(M,g^{+})$. Since
this primarily reflects features of the underlying manifold [23] (not of the
Wick rotation) we leave this unexplored here.

The proof of this theorem uses the following lemmas.

######  Lemma 4.3.

For all $(\zeta,y)\in\Sigma_{\tilde{\theta}}\times M$ there exists a unique
$\kappa_{\zeta,y}^{\theta}\in L^{2}(M)$ such that for every $\psi\in L^{2}(M)$

$\displaystyle(e^{\zeta\Delta_{\theta}}\psi)(y)=\left\langle\kappa_{\zeta,y}^{\theta}\,\middle|\,\psi\right\rangle_{L^{2}(M)}\,,\quad\forall\,(\zeta,y)\in\Sigma_{\tilde{\theta}}\times
M\,.$ (4.17)

Proof of Lemma 4.3. Fixing an arbitrary
$(\zeta,y)\in\Sigma_{\tilde{\theta}}\times M$, consider the mapping
$L^{2}(M)\ni\psi\mapsto(e^{\zeta\Delta_{\theta}}\psi)(y)\in{\mathbb{C}}$,
which is well-defined by Theorem 4.1 and clearly linear. Moreover, Theorem
4.1(ii) implies there is a constant $B>0$ (depending on $y,\zeta,\theta$, but
independent of $\psi\in L^{2}(M)$) such that

$\displaystyle\big{|}(e^{\zeta\Delta_{\theta}}\psi)(y)\big{|}\leq
B\norm{\psi}_{L^{2}(M)}\,,\quad\forall\,\psi\in L^{2}(M)\,.$ (4.18)

Thus, this mapping is continuous, and by the Riesz Representation Theorem is
realized by a unique $\kappa_{\zeta,y}^{\theta}\in L^{2}(M)$ via
$(e^{\zeta\Delta_{\theta}}\psi)(y)=\langle\kappa_{\zeta,y}^{\theta}|\psi\rangle_{L^{2}(M)}$,
for each $\psi\in L^{2}(M)$. ∎

Next, as a tool to prove the regularity statement Theorem 4.2(ii), we recall
the notions of weak and strong differentiability of Hilbert space valued maps.
Let ${\mathcal{O}}\subseteq{\mathbb{R}}^{n}$ be an open set and
$h:{\mathcal{O}}\to L^{2}(M)$.

  * •

$h$ is said to be weakly $C^{k},\,k\in{\mathbb{N}}_{0}$, if
$\forall\,\varphi\in
L^{2}(M):\,\left\langle\varphi\,\middle|\,h(\cdot)\right\rangle_{L^{2}(M)}$ is
in $C^{k}({\mathcal{O}})$.

  * •

The Gâteaux derivative of $h$ along direction $\hat{e}_{i}$ in $\Omega$ is

$\displaystyle{\partial}_{i}h(z):=\lim_{\varepsilon\to
0}\frac{h(z+\varepsilon\hat{e}_{i})-h(z)}{\varepsilon}\,,$ (4.19)

with the limit taken in $\norm{\cdot}_{L^{2}(M)}$. Then $h$ is said to be
strongly $C^{k},\,k\in{\mathbb{N}}_{0}$, if for all multi-indices $\alpha$
with $|\alpha|\leq k$ the Gâteaux derivatives ${\partial}^{\alpha}h$ exist
throughout $\Omega$ and are strongly continuous.

Then we have the result

######  Lemma 4.4.

$h:\Omega\to L^{2}(M)$ is strongly $C^{\infty}$ iff it is weakly $C^{\infty}$.

We omit the proof, referring to Corollary 1.42 of [9].

We now proceed to the proof of Theorem 4.2.

Proof of Theorem 4.2. We begin with the existence of the pointwise defined
integral kernel. Fixing arbitrary
$\zeta_{1},\zeta_{2}\in\Sigma_{\tilde{\theta}}$, $y\in M$ and $\psi\in
L^{2}(M)$, consider

$\displaystyle(e^{(\zeta_{1}+\zeta_{2})\Delta_{\theta}}\psi)(y)$
$\displaystyle=\big{(}e^{\zeta_{1}\Delta_{\theta}}(e^{\zeta_{2}\Delta_{\theta}}\psi)\big{)}(y)=\expectationvalue{\kappa_{\zeta_{1},y}^{\theta}\,\big{|}\,e^{\zeta_{2}\Delta_{\theta}}\psi}_{L^{2}(M)}=\left\langle
e^{\zeta_{2}^{\ast}\Delta_{\pi-\theta}}\kappa^{\theta}_{\zeta_{1},y}\,\middle|\,\psi\right\rangle_{L^{2}(M)}$
$\displaystyle=\int\\!d\mu_{g}(y^{\prime})\,(e^{\zeta_{2}^{\ast}\Delta_{\pi-\theta}}\kappa^{\theta}_{\zeta_{1},y})^{\ast}(y^{\prime})\psi(y^{\prime})$
$\displaystyle=\int\\!d\mu_{g}(y^{\prime})\,\left\langle\kappa_{\zeta_{1},y}^{\theta}\,\middle|\,\kappa_{\zeta_{2}^{\ast},y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}\psi(y^{\prime})\,,$
(4.20)

where we use
$e^{\zeta_{2}^{\ast}\Delta_{\pi-\theta}}=(e^{\zeta_{2}\Delta_{\theta}})^{\ast}$
for the third equality on the first line, and Lemma 4.3 for the last step. In
particular, for $\zeta\in\Sigma_{\tilde{\theta}}$ we have

$\displaystyle(e^{\zeta\Delta_{\theta}}\psi)(y)$
$\displaystyle=\int\\!d\mu_{g}(y^{\prime})\,\left\langle\kappa_{\zeta/2,y}^{\theta}\,\middle|\,\kappa_{\zeta^{\ast}/2,y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}\psi(y^{\prime})\,,$
(4.21)

so the semigroup has a pointwise defined integral kernel

$\displaystyle
K_{\zeta}^{\theta}(y,y^{\prime}):=\left\langle\kappa_{\zeta/2,y}^{\theta}\,\middle|\,\kappa_{\zeta^{\ast}/2,y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}\,,\quad\forall\,(\zeta,y,y^{\prime})\in\Sigma_{\tilde{\theta}}\times
M\times M\,.$ (4.22)

(i) The hermiticity property of the kernel follows immediately from the
definition (4.22).

(ii) We now turn to its regularity. Let ${\mathcal{O}}$ be a chart of
$\Sigma_{\tilde{\theta}}\times M$ with coordinates $(\zeta,y)$, and consider
the map $(\zeta,y)\mapsto\kappa^{\theta}_{\zeta,y}\in L^{2}(M)$. Since
$\langle\kappa^{\theta}_{\zeta,y}|\psi\rangle_{L^{2}(M)}=(e^{\zeta\Delta_{\theta}}\psi)(y)$
is smooth in $(\zeta,y)\in{\mathcal{O}}$ for every $\psi\in L^{2}(M)$, the map
is weakly $C^{\infty}$, and hence strongly $C^{\infty}$ by the Lemma 4.4.
Similarly, choosing another chart ${\mathcal{O}}^{\prime}$ of
$\Sigma_{\tilde{\theta}}\times M$ with coordinates $(\xi,y^{\prime})$, define
the map

$\displaystyle{\mathcal{O}}\times{\mathcal{O}}^{\prime}\ni(\zeta,y;\xi,y^{\prime})\mapsto\left\langle\kappa_{\zeta,y}^{\theta}\,\middle|\,\kappa_{\xi^{\ast},y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}\in{\mathbb{C}}\,.$
(4.23)

This is jointly smooth in $(\zeta,y;\xi,y^{\prime})$ since the maps
$(\zeta,y)\mapsto\kappa^{\theta}_{\zeta,y}$ and
$(\xi,y^{\prime})\mapsto\kappa^{\pi-\theta}_{\xi^{\ast},y^{\prime}}$ are
strongly $C^{\infty}$. Since the charts were arbitrary,
$(\Sigma_{\tilde{\theta}}\times
M)^{2}\ni(\zeta,y;\xi,y^{\prime})\mapsto\langle\kappa^{\theta}_{\zeta,y}|\kappa_{\xi^{\ast},y^{\prime}}^{\pi-\theta}\rangle_{L^{2}(M)}$
is smooth. Restricting to the diagonal in kernel time yields joint
$C^{\infty}$-smoothness of

$\displaystyle\Sigma_{\tilde{\theta}}\times M\times
M\ni(\zeta,y,y^{\prime})\mapsto\left\langle\kappa_{\zeta/2,y}^{\theta}\,\middle|\,\kappa_{\zeta^{\ast}/2,y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}=K_{\zeta}^{\theta}(y,y^{\prime})\in{\mathbb{C}}\,.$
(4.24)

Finally, the analyticity in $\zeta$ is a by-product of Theorem 4.1(i) and
equation (4.26) below.

(iii) Next, fixing $\zeta\in\Sigma_{\tilde{\theta}}$, the uniqueness of the
kernel $K_{\zeta}^{\theta}\in C^{\infty}(M\times M)$ is implied by the
uniqueness (for each $y\in M$) of $\kappa_{\zeta,y}^{\theta}\in L^{2}(M)$ from
Lemma 4.3. In more detail, fixing $y\in M$, and comparing the realizations of
$(e^{\zeta\Delta_{\theta}}\psi)(y)$ in terms of $\kappa_{\zeta,y}^{\theta}$
via Lemma 4.3 and the kernel $K_{\zeta}^{\theta}$ in (4.12), yields
$\kappa_{\zeta,y}^{\theta}(y^{\prime})^{\ast}=K_{\zeta}^{\theta}(y,y^{\prime})$
for a.e. $y^{\prime}\in M$. The same must be true of any other kernel
$J_{\zeta}^{\theta}\in C^{\infty}(M\times M)$ satisfying (4.12), i.e. for each
$y\in M$, they coincide a.e. $y^{\prime}\in M$. Then smoothness implies they
coincide everywhere, i.e. $J_{\zeta}^{\theta}=K_{\zeta}^{\theta}$.

(iv) To prove the semigroup identity, fix arbitrary
$\zeta\in\Sigma_{\tilde{\theta}}$ and $y,y^{\prime}\in M$. As discussed in
(iii) above, for any $z\in
M:\kappa_{\zeta,z}^{\theta}(z^{\prime})^{\ast}=K_{\zeta}^{\theta}(z,z^{\prime})$
for a.e. $z^{\prime}\in M$. Then, it is sufficient to prove that for
$\zeta_{1},\zeta_{2}\in\Sigma_{\tilde{\theta}}$

$\displaystyle\forall\,y,y^{\prime}\in
M:\,\,K_{\zeta_{1}+\zeta_{2}}^{\theta}(y,y^{\prime})=\left\langle\kappa_{\zeta_{1},y}^{\theta}\,\middle|\,\kappa_{\zeta_{2}^{\ast},y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}\,.$
(4.25)

Indeed, from the definition of the kernel (4.21) and the computation (4), it
is immediate that for each $y\in
M:\,K_{\zeta_{1}+\zeta_{2}}^{\theta}(y,y^{\prime})=\langle\kappa_{\zeta_{1},y}^{\theta}|\kappa_{\zeta_{2}^{\ast},y^{\prime}}^{\pi-\theta}\rangle_{L^{2}(M)}$
a.e. $y^{\prime}\in M$. However, both sides are smooth in $y^{\prime}\in M$ by
the above regularity results (4.23), (4.24), and hence are equal for all
$y^{\prime}\in M$. This proves (4.25), and hence the semigroup property.

(v) To see that for each $y^{\prime}\in M$,
$({\partial}_{\zeta}-\Delta_{\theta,y})K_{\zeta}^{\theta}(y,y^{\prime})=0$ in
$\Sigma_{\tilde{\theta}}\times M$, fix an arbitrary
$\xi\in\Sigma_{\tilde{\theta}}$ such that
$\zeta-\xi\in\Sigma_{\tilde{\theta}}$ and consider
$f(\zeta,y):=\langle\kappa_{\zeta,y}^{\theta}|\kappa_{\xi^{\ast},y^{\prime}}^{\pi-\theta}\rangle_{L^{2}(M)}=(e^{\zeta\Delta_{\theta}}\kappa_{\xi^{\ast},y^{\prime}}^{\pi-\theta})(y)$.
Then Theorem 4.1(i) implies
$({\partial}_{\zeta}-\Delta_{\theta})f(\zeta,y)=0$, and hence the same holds
for $f(\zeta-\xi,y)$ since $\zeta-\xi\in\Sigma_{\tilde{\theta}}$. The
semigroup identity (4.25) entails that

$\displaystyle
f(\zeta-\xi,y)=\left\langle\kappa_{\zeta-\xi,y}^{\theta}\,\middle|\,\kappa_{\xi^{\ast},y^{\prime}}^{\pi-\theta}\right\rangle_{L^{2}(M)}=K_{\zeta}^{\theta}(y,y^{\prime})\,,$
(4.26)

yielding (4.15). Finally, the limit statement (4.16) is an immediate corollary
of Theorem 4.1(iii), completing the proof. ∎

## 5\. Asymptotic expansion of diagonal Wick rotated heat kernel

Much of the computational uses of the heat kernel rest on it’s asymptotic
expansion for small diffusion time. The tabulated low order coefficients
provide a shortcut to many otherwise difficult quantum field theoretical
computations on curved backgrounds. The goal in the following is to establish
Theorem 5.6, a generalization of this expansion for the Wick rotated heat
kernel.

For the heat kernel proper the off-diagonal expansion is usually formulated in
terms of the Synge function (one-half of the square of the geodesic distance).
While the Synge function is a natural geometric quantity it is itself
nontrivial to construct, both mathematically [7, 35] and computationally. In
our Wick rotated setting the standard local existence proofs carry over
straightforwardly only if the metric is assumed to be locally analytic. This
is somewhat at odds with our framework and presumably also not a necessary
assumption. We therefore postpone a detailed investigation of the Wick rotated
Synge function to another occasion and proceed here along different lines. We
begin by noting the correctly Wick rotated form of the Eikonal equation

$\sigma_{\theta}(y,y^{\prime})=\frac{ie^{-i\theta}}{2}g_{\theta}^{\mu\nu}(y){\partial}_{\mu}\sigma_{\theta}{\partial}_{\nu}\sigma_{\theta}\,,$
(5.1)

which characterizes the Synge function $\sigma_{\theta}$ subject to suitable
boundary conditions. We omit a derivation as (5.1) will occur later as a
byproduct of the ansatz for $K_{\zeta}^{\theta}(y,y^{\prime})$ used. Note that
(5.1) is not symmetric in $y,y^{\prime}$ while the solution aimed at is. The
relevant boundary conditions are $\sigma_{\theta}(y,y)=0$,
$\nabla^{\theta}_{\mu}\nabla^{\theta}_{\nu}\sigma_{\theta}|_{y=y^{\prime}}=-ie^{i\theta}g^{\theta}_{\mu\nu}$,
where $\nabla_{\mu}^{\theta}$ is the metric connection build from
$g^{\theta}$. The usual construction reduces (for $\theta=0,\pi/2$) the
solution of (5.1) to the solution of the geodesic equation. In the present
context this would require evaluation of the metric at complex arguments,
which is ambiguous without local analyticity. In the following we instead use
an explicitly constructable asymptotic solution.

######  Proposition 5.1.

Let $g^{\theta}$ be a Wick rotated smooth metric on $M$ and fix a local chart
$U\subset M$, which in local coordinates can be identified with an open subset
of $\mathbb{R}^{d+1}$. For $y,y^{\prime}\in U$ set ${\cal
E}_{\theta}[\sigma](y,y^{\prime}):=\sigma(y,y^{\prime})-\frac{i}{2}e^{-i\theta}g_{\theta}^{\mu\nu}(y){\partial}_{\mu}\sigma{\partial}_{\nu}\sigma$,
as well as $\Delta y^{\mu}:=(y-y^{\prime})^{\mu}$ and $|\Delta
y|:=(\delta_{\mu\nu}\Delta y^{\mu}\Delta y^{\nu})^{1/2}$.

  * (a)

Inserting the formal series

$\sigma_{\theta}(y,y^{\prime})=-ie^{i\theta}\sum_{n\geq
2}\frac{1}{n!}s_{\mu_{1}\ldots\mu_{n}}(y)\Delta y^{\mu_{1}}\ldots\Delta
y^{\mu_{n}}\,,$ (5.2)

into ${\cal E}_{\theta}[\sigma]\equiv 0$, the completely symmetric
coefficients $s(y)$ are uniquely determined differential polynomials in
$g_{\mu\nu}^{\theta}$ contracted with $g_{\theta}^{\rho\sigma}$ at $y$.

  * (b)

There exists a (possibly non-unique) smooth function
$\tilde{\sigma}_{\theta}(y,y^{\prime})$, which in a $\Delta y$ expansion has
Taylor coefficients $-ie^{i\theta}s_{\mu_{1}\ldots\mu_{n}}(y)$, such for all
$N\geq 1$

$\big{|}{\cal E}_{\theta}[\tilde{\sigma}_{\theta}](y,y^{\prime})\big{|}\leq
C_{N}|\Delta y|^{N}\,,$ (5.3)

for some constants $C_{N}>0$ and all $y,y^{\prime}$ in an open neighborhood
$V\subset U$.

  * (c)

There exists $\theta$ dependent constants $0<c_{-}\leq c_{+}<\infty$ and an
open neighborhood $V\subset U$ such that for all $y,y^{\prime}\in V$

$\displaystyle c_{-}|\Delta y|^{2}\leq{\rm
Re}\,\tilde{\sigma}_{\theta}(y,y^{\prime})\leq c_{+}|\Delta y|^{2}\,.$ (5.4)

Proof. (a) Technically, it is convenient to insert an ansatz akin to (5.2) but
with coefficients evaluated at $y^{\prime}$ into into ${\cal E}[\sigma]=0$. By
comparing powers of $\Delta y$ then only algebraic recursion relations arise.
Moreover, these coefficients at $y^{\prime}$ are differential polynomials in
$g^{\theta}_{\mu\nu}$ with at most $n-2$ differentiations at order $n$,
contracted with $g^{\rho\sigma}_{\theta}$. By Taylor expanding the
coefficients at $y^{\prime}=y-\Delta y$ around $y$ the expansion (5.2) arises,
where each $s(y)$ coefficient is a finite linear combination of the
algebraically determined coefficients at $y^{\prime}$.

(b) By the assumed smoothness of the metric there exist a precompact
$V_{1}\subset U$ such that $g_{\mu\nu}^{\theta}$, $g_{\theta}^{\mu\nu}$, and
all its derivatives are bounded on $V_{1}$. By (a) the coefficients $s(y)$ can
be bounded in terms of the Taylor coefficients of the metric at $y$. For
arbitrary $N\geq 2$, the remainders of the partial sums, $n\leq N\\!-\\!1$,
can thus be bounded by a constant times a $O(\Delta y^{N})$ contraction. For
some open $V\subset V_{1}$ this gives a bound $C_{N}|\Delta y|^{N}$, for all
$y,y^{\prime}\in V$. Borel’s lemma then provides the existence of
$\tilde{\sigma}_{\theta}$ satisfying (5.3).

(c) We fix again some precompact set $V_{1}\subset U$ in which the components
of $g_{\mu\nu}^{\theta}$, $g_{\theta}^{\mu\nu}$, and all derivatives thereof
are bounded. Defining

$\displaystyle c_{-}:=\frac{1}{2}\inf_{y\in V_{1}}\inf_{\Delta
y\in\mathbb{R}^{d+1},|\Delta y|=1}{\rm
Re}\Big{(}\\!-ie^{i\theta}g_{\mu\nu}^{\theta}(y)\Delta y^{\mu}\Delta
y^{\nu}\Big{)}\,,$ $\displaystyle c_{+}:=\frac{1}{2}\sup_{y\in
V_{1}}\sup_{\Delta y\in\mathbb{R}^{d+1},|\Delta y|=1}{\rm
Re}\Big{(}\\!-ie^{i\theta}g_{\mu\nu}^{\theta}(y)\Delta y^{\mu}\Delta
y^{\nu}\Big{)}\,,$ (5.5)

one has $0<c_{-}<c_{+}<\infty$. The function $V_{1}\times
V_{1}\ni(y,y^{\prime})\mapsto\tilde{\sigma}_{\theta}(y,y^{\prime})-c_{-}|\Delta
y|^{2}$ is smooth and vanishes on the diagonal. By (a), (b) one also has
$2{\rm Re}\tilde{\sigma}_{\theta}(y,y^{\prime})={\rm
Re}(\\!-ie^{i\theta}g_{\mu\nu}^{\theta}\Delta y^{\mu}\Delta y^{\nu})+O(|\Delta
y|^{3})$ near the diagonal. It follows that ${\rm
Re}\,\tilde{\sigma}_{\theta}(y,y^{\prime})-c_{-}|\Delta y|^{2}>0$ in some open
neighborhood of the diagonal in $V_{1}\times V_{1}$. This implies that there
is an open neighborhood $V\subset V_{1}$ such that ${\rm
Re}\,\tilde{\sigma}_{\theta}(y,y^{\prime})>c_{-}|\Delta y|^{2}$, for all
$y,y^{\prime}\in V$. The upper bound in the first equation of (5.4) is
obtained analogously. ∎

To low orders the coefficients read

$\displaystyle s_{\mu_{1}\mu_{2}}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
g^{\theta}_{\mu_{1}\mu_{2}}\,,\quad
s_{\mu_{1}\mu_{2}\mu_{3}}=-\frac{3}{2}{\partial}_{\mu_{1}}g^{\theta}_{\mu_{2}\mu_{3}}\,,$
$\displaystyle s_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle
2{\partial}_{\mu_{1}}{\partial}_{\mu_{2}}g^{\theta}_{\mu_{3}\mu_{4}}-\frac{1}{4}g_{\theta}^{\rho\sigma}{\partial}_{\rho}g^{\theta}_{\mu_{1}\mu_{2}}{\partial}_{\sigma}g^{\theta}_{\mu_{3}\mu_{4}}$
(5.6) $\displaystyle+$ $\displaystyle
g_{\theta}^{\rho\sigma}{\partial}_{\rho}g^{\theta}_{\mu_{1}\mu_{2}}{\partial}_{\mu_{3}}g^{\theta}_{\sigma\mu_{4}}-g_{\theta}^{\rho\sigma}{\partial}_{\mu_{1}}g^{\theta}_{\rho\mu_{2}}{\partial}_{\mu_{3}}g^{\theta}_{\sigma\mu_{4}}\,,$

where symmetrization in all indices is understood. An analogous expansion
exists with coefficients $\mathfrak{s}_{\mu_{1}\ldots\mu_{n}}$ evaluated at
$y^{\prime}$ rather than $y$. A-posteriori one then finds
$\mathfrak{s}_{\mu_{1}\ldots\mu_{n}}=(-)^{n}s_{\mu_{1}\ldots\mu_{n}}$, $n\geq
2$, indicating that the directly computed coefficients are compatible with the
symmetry of the solution,
$\sigma_{\theta}(y,y^{\prime})=\sigma_{\theta}(y^{\prime},y)$. For the present
purposes symmetry of the truncated series (5.2) will not be needed. The
following proposition summarizes a Wick rotated version of the usual starting
point to establish the existence of an asymptotic expansion.

######  Proposition 5.2.

Let $\sigma_{\theta}(y,y^{\prime})$ be an exact, symmetric solution of the
Eikonal equation (5.1). Then

$\displaystyle
K_{\zeta}^{\theta}(y;y^{\prime})=\frac{(-ie^{i\theta})^{\frac{d-1}{2}}}{(4\pi\zeta)^{\frac{d+1}{2}}}e^{-\frac{1}{2\zeta}\sigma_{\theta}(y,y^{\prime})}\sum_{n\geq
0}A_{n}^{\theta}(y,y^{\prime})(ie^{-i\theta}\zeta)^{n},$ (5.7)

provides a formal solution to the heat equation
$({\partial}_{\zeta}-\Delta_{\theta})K^{\theta}_{\zeta}=0$, iff the
coefficients $A_{n}^{\theta}$ satisfy

$\displaystyle
2g_{\theta}^{\mu\nu}{\partial}_{\mu}\sigma_{\theta}{\partial}_{\nu}A_{0}^{\theta}+[\nabla_{\theta}^{2}\sigma_{\theta}+ie^{i\theta}(d\\!+\\!1)]A_{0}^{\theta}=0\,,$
(5.8) $\displaystyle
2g_{\theta}^{\mu\nu}{\partial}_{\mu}\sigma_{\theta}{\partial}_{\nu}A_{n+1}^{\theta}+\big{[}\nabla_{\theta}^{2}\sigma_{\theta}-\frac{ie^{i\theta}}{2}(2n\\!+\\!1\\!-\\!d)\big{]}A_{n+1}^{\theta}=-e^{2i\theta}\Delta_{\theta}A_{n}^{\theta}\,,\quad
n\geq 0\,.$

The overall normalization is fixed by $A_{0}(y,y)=1$. Further,
$\nabla_{\mu}^{\theta}$ is the metric connection associated with $g^{\theta}$,
and
$\nabla_{\theta}^{2}=g^{\mu\nu}_{\theta}\nabla^{\theta}_{\mu}\nabla^{\theta}_{\nu}$.

Proof. Direct substitution and comparing powers of $\zeta$. ∎

The recursion (LABEL:hkexp2) generalizes the well-known one for the off-
diagonal heat kernel coefficients $A_{n}$ (see e.g. [18], Eq. 9.9a), and
specializes to them for $\theta=\pi/2$. For Euclidean signature $A_{0}$ is
expressible in terms of the Van Vleck-Morette determinant and the recursion
can be solved via the method of characteristics. The construction masks the
symmetry of the coefficients, which has to be established independently [34].

Even with the symmetry of the coefficients ensured, the Euclidean signature
series in (5.7) may not be properly asymptotic to the exact heat kernel [34].
A complete and direct proof for the existence of a small time asymptotic
expansion for the off-diagonal heat kernel has only been given recently in
[32], building on [28]. The methodology in [32, 28] does not make use of the
transport equations (LABEL:hkexp2) and is therefore somewhat decoupled from
the way the (Euclidean signature version of the) ansatz (5.7) enters
computations. It is also not immediate that the methodology would carry over
to the Wick rotated situation. Since in applications mostly the diagonal
expansion is needed, we aim in the following only at a small $\zeta$
asymptotic expansion for the diagonal of our kernel
$K_{\zeta}^{\theta}(y,y^{\prime})$. By modifying a proof strategy proposed by
Wald [47], and approximating the solutions of the transport equations via
Proposition 5.1 a fairly elementary proof for the existence of the diagonal
expansion can be given that retains contact to (5.7), (LABEL:hkexp2). This is
possible because we have independently established the existence of our kernel
$K_{\zeta}^{\theta}(y,y^{\prime})$ with the requisite properties. In contrast,
the approaches [32, 28, 22, 47] aim at establishing the kernel’s existence
through a sequence of approximants that are asymptotic a-posteriori.

Local parametrix. For simplicity, we take the heat kernel time $\zeta$ in the
following to be real (and positive) and comment on the extension of Theorem
5.6 below to $|{\rm Arg}\zeta|<\tilde{\theta}$ at the end. Given an arbitrary
$y_{0}\in M$, an open chart neighborhood $U\ni y_{0}$, and a truncation order
$N\in{\mathbb{N}}$, we initially consider a local parametrix defined in
$U\times U$ of the form

$\displaystyle
F_{\zeta}(y,y^{\prime}):=\frac{(-ie^{i\theta})^{\frac{d-1}{2}}}{(4\pi\zeta)^{\frac{d+1}{2}}}e^{-\frac{1}{2\zeta}s_{\theta}(y,y^{\prime})}\sum_{n=0}^{N}A_{n}^{\theta}(y,y^{\prime})(ie^{-i\theta}\zeta)^{n}\,.$
(5.9)

Later on $s_{\theta}$ will be identified with the function
$\tilde{\sigma}_{\theta}(y,y^{\prime})$ from Proposition 5.1; for now we only
need $s_{\theta}(y,y^{\prime})$ to be jointly smooth in $\zeta,y,y^{\prime}$,
for $y,y^{\prime}\in U$ and such that $\real s_{\theta}>0$ in a neighborhood
of $y=y^{\prime}$. Importantly, $s_{\theta}$ is neither necessarily symmetric
nor a solution of the Eikonal equation (5.1). Similarly, the $A_{n}^{\theta}$
are initially only assumed to be smooth in $y,y^{\prime}\in U$ and nonzero in
a neighborhood of $y=y^{\prime}$. To unclutter the notation we suppress the
$N$ and $\theta$ dependence in the notation.

To proceed, we act with the heat operator ${\partial}_{\zeta}-\Delta_{\theta}$
on (5.9). For convenience we recall the definitions,
$\Delta_{\theta}=ie^{-i\theta}(\nabla_{\theta}^{2}-V)$,
$\nabla_{\theta}^{2}=g_{\theta}^{\mu\nu}\nabla^{\theta}_{\mu}\nabla^{\theta}_{\nu}$.
A straightforward computation gives

$\displaystyle\\!\\!\\!\big{[}{\partial}_{\zeta}-\Delta_{\theta,y}\big{]}F_{\zeta}(y,y^{\prime})$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{(-ie^{i\theta})^{\frac{d-1}{2}}}{(4\pi\zeta)^{\frac{d+1}{2}}}e^{-\frac{1}{2\zeta}s_{\theta}(y,y^{\prime})}$
(5.10) $\displaystyle\times$
$\displaystyle\sum_{n=0}^{N}(ie^{-i\theta}\zeta)^{n}\Big{[}\frac{X_{n}}{(ie^{-i\theta}\zeta)^{2}}+\frac{Y_{n}}{ie^{-i\theta}\zeta}+Z_{n}\Big{]}\,,$

where

$\displaystyle X_{n}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{2}(ie^{-i\theta})^{2}\Big{(}s_{\theta}-\frac{1}{2}ie^{-i\theta}g_{\theta}^{\mu\nu}{\partial}_{\mu}s_{\theta}{\partial}_{\nu}s_{\theta}\Big{)}A_{n}^{\theta}\,,$
$\displaystyle Y_{n}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{2}ie^{-i\theta}\big{[}(2n-(d\\!+\\!1))\mbox{11}+{\mathcal{L}}(s_{\theta})\big{]}A_{n}^{\theta}\,,\quad
Z_{n}=-\Delta_{\theta}A_{n}^{\theta}\,,$
$\displaystyle{\mathcal{L}}(s_{\theta})$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle
ie^{-i\theta}\nabla^{2}_{\theta}s_{\theta}+2ie^{-i\theta}g_{\theta}^{\mu\nu}{\partial}_{\mu}s_{\theta}{\partial}_{\nu}\,.$
(5.11)

The structure of the $X_{n}$ term elucidates the origin of the Eikonal
equation (5.1). Clearly, $X_{n}=0$ for all $n\geq 0$ iff (for at least one
nonzero $A_{n_{0}}^{\theta}$) $s_{\theta}=\sigma_{\theta}$ solves (5.1).
Further, for $X_{n}=0$, $n\geq 0$, the remaining terms in the second line
reduce to $(ie^{-i\theta}\zeta)^{N}Z_{N}$ iff $Y_{0}=0$ and $Y_{n+1}+Z_{n}=0$,
$n=0,\ldots,N\\!-\\!1$. These vanishing conditions are precisely the
$n=0,\ldots,N\\!-\\!1$ transport equations in (LABEL:hkexp2).

The strategy used later on will be to impose approximate vanishing conditions
for the $X_{n}$, $n\geq 0$, and $Y_{n+1}+Z_{n},n=0,\ldots,N\\!-\\!1$, in an
expansion in powers of $\Delta y$. This needs to be done in a way that is
coordinated with the order of the expansion in $\zeta$. To do so, we interpret
(and eventually construct) the functions
$s_{\theta}(y,y^{\prime}),A_{n}^{\theta}(y,y^{\prime}),F_{\zeta}(y,y^{\prime})$
as truncated series in $\Delta y$ with coefficients evaluated at $y$. Our
convention throughout this section will be that the first argument indicates
the base point at which the coefficients are evaluated,
$f(y,y^{\prime})=f(y)+f_{\mu}(y)\Delta y^{\mu}+O(|\Delta y|^{2})$, etc.. The
remainders in these expansions will be of $O(|\Delta y|^{L})$, for suitable
powers $L$. For the next steps, we assume only that $s_{\theta}$ has a leading
term given by the one fixed by Proposition 5.1, i.e.

$s_{\theta}(y,y^{\prime})=-\frac{i}{2}e^{i\theta}g^{\theta}_{\mu\nu}(y)\Delta
y^{\mu}\Delta y^{\nu}+O(|\Delta y|^{3})\,,$ (5.12)

that $A_{n}^{\theta}(y,y^{\prime})$ is regular and nonzero as $|\Delta
y|\rightarrow 0$, and in particular $A_{0}^{\theta}(y,y)=1$. Before turning to
the $\Delta y$ expansion, we prepare a key property of the local parametrix.
All function spaces occurring in the following are subspaces of $L^{2}(M)$ and
$\|\cdot\|$ will always refer to the $L^{2}(M)$ norm.

######  Lemma 5.3.

Let $F_{\zeta}(y,y)$, $\zeta>0$, be the local parametrix (5.9) defined in an
open chart neighborhood $U$ of the point $y_{0}\in M$. Then there is a
precompact open neighborhood $V\Subset U$ of $y_{0}$ such that :

  * (a)

As a distribution on $C^{\infty}_{c}(V)$ test functions,

$\lim_{\zeta\rightarrow
0^{+}}F_{\zeta}(y,y^{\prime})=|g|^{-1/2}(y)\,\delta(y-y^{\prime})\,.$ (5.13)

  * (b)

Let $\mathbb{F}_{\zeta}$, $\zeta>0$, be the integral operator on $L^{2}(V)$
defined by the kernel $F_{\zeta}$. Then

$\lim_{\zeta\rightarrow
0^{+}}\|\mathbb{F}_{\zeta}\psi\|=\|\psi\|\,,\quad\psi\in L^{2}(V)\,.$ (5.14)

Proof. (a) It follows from (5.12) that $y_{0}$ has a precompact open
neighborhood $V\Subset U$ where
$|2s_{\theta}(y,y^{\prime})+ie^{i\theta}g_{\mu\nu}(y)\Delta y^{\mu}\Delta
y^{\nu}|<c_{-}|\Delta y|^{2}$, for all $y,y^{\prime}\in V$. As in the proof of
Proposition 5.1(c) this implies

$\frac{c_{-}}{4}|\Delta y|^{2}\leq\frac{1}{2\zeta}{\rm
Re}\big{[}s_{\theta}\big{(}y,y\\!-\\!\sqrt{\zeta}\Delta y\big{)}\big{]}\,,$
(5.15)

for all $y,y^{\prime}\in V$, as needed shortly. We express (5.13) in terms of
the integral operator ${\mathbb{F}}_{\zeta}$ with kernel
$F_{\zeta}(y,y^{\prime})$. Changing integration variables from $y^{\prime}$ to
$\Delta y$, its action on a $C_{c}^{\infty}(V)$ function $\psi$ can be written
as

$\displaystyle({\mathbb{F}}_{\zeta}\psi)(y)=\frac{(-ie^{i\theta})^{\frac{d-1}{2}}}{(4\pi\zeta)^{\frac{d+1}{2}}}\sum_{n=0}^{N}(ie^{-i\theta}\zeta)^{n}\\!\int\\!d^{d+1}(\Delta
y)f_{\zeta,n}(y,\Delta y)\,,$ (5.16) $\displaystyle f_{\zeta,n}(y,\Delta
y)=|g|^{1/2}(y\\!-\\!\Delta y)e^{-\frac{1}{2\zeta}s_{\theta}(y,y-\Delta
y)}A_{n}^{\theta}(y,y\\!-\\!\Delta y)\psi(y\\!-\\!\Delta y)\,.$

Next, we rescale the integration variable $\Delta y\mapsto\zeta^{1/2}\Delta
y$. In the prefactor before the sum this cancels the $\zeta^{-\frac{d+1}{2}}$,
leaving only non-negative $\zeta$ powers in front of the $d^{d+1}\Delta y$
measure. The precompactness of $V$ entails that the integration ranges over a
bounded domain, say $|\Delta y|\leq\delta$. After the rescaling one has
$|\Delta y|\leq\delta/\sqrt{\zeta}$, so for $\zeta\rightarrow 0^{+}$ the
integration will effectively extend over all of $\mathbb{R}^{d+1}$. Since the
$A_{n}^{\theta}$ are smooth on $U\times U$, each may be bounded by its
supremum over the precompact $V\times V$. Then (5.15) can be used to bound for
each $n$ the integrand by a constant times $\exp\\{-c_{-}|\Delta y|^{2}/4\\}$,
which is of course integrable. Applying dominated convergence the
$\zeta\rightarrow 0^{+}$ limit can be brought inside the integral. Pointwise
in $\Delta y$ the $n$-th integrand converges to

$|g|^{1/2}(y)A_{n}(y,y)\psi(y)\exp\\{\frac{i}{4}e^{i\theta}g^{\theta}_{{\mu\nu}}(y)\Delta
y^{\mu}\Delta y^{\nu}\Big{\\}}\,.$ (5.17)

The $d^{d+1}\Delta y$ integral is a well-defined Gaussian, as
$\theta\in(0,\pi)$. Performing it cancels the prefactors of the
$\sum_{n=0}^{N}(ie^{-i\theta}\zeta)^{n}$ sum. When taking the
$\zeta\rightarrow 0^{+}$ limit in (LABEL:lim4) only the $n=0$ term in the sum
contributes and one arrives at $\lim_{\zeta\rightarrow
0^{+}}({\mathbb{F}}_{\zeta}\psi)(y)=\psi(y)$, using $A^{\theta}_{0}(y,y)=1$.
This gives (5.13) as a distribution on $C^{\infty}_{c}(V)$ test functions.

(b) Using (LABEL:lim4) and rescaling both difference integration variables
gives

$\displaystyle\int\\!dy\,|g|^{1/2}(y)\,|\mathbb{F}_{\zeta}\psi|(y)^{2}=\frac{1}{(4\pi)^{d+1}}\sum_{m,n=0}^{N}\zeta^{n+m}\int\\!d^{d+1}y\,|g|^{1/2}(y)\int\\!d^{d+1}\Delta
y\int\\!d^{d+1}\Delta z$ $\displaystyle\quad\times
f_{\zeta,m}(y,\sqrt{\zeta}\Delta z)^{*}f_{\zeta,n}(y,\sqrt{\zeta}\Delta y)\,.$
(5.18)

The integration in $y$ extends over a bounded domain $V$, while after the
rescaling the integration domain over the difference variables $\Delta y$,
$\Delta z$ will tend to $\mathbb{R}^{d+1}$, as $\zeta\rightarrow 0^{+}$.
Pointwise in $\Delta y$, $\Delta z$, the integrand converges for
$\zeta\rightarrow 0^{+}$ to the product of (5.17) and a complex conjugate copy
with $n\mapsto m,\Delta y\mapsto\Delta z$. By an argument analogous to that in
the proof of part (a) above, the integrand is bounded for each $n,m$ by a
constant times

$|g|^{1/2}(y)|\psi(y)|^{2}\exp\\{-c_{-}|\Delta y|^{2}/4\\}\exp\\{-c_{-}|\Delta
z|^{2}/4\\}\,.$ (5.19)

This is because $\psi$ has compact support, so for fixed $y$ the
$\psi(y-\sqrt{\zeta}\Delta y)$ term can be bounded by its finite supremum over
$\Delta y\in\mathbb{R}^{d+1}$, and similarly for $\psi(y-\sqrt{\zeta}\Delta
z)^{*}$. Since the $y$ integration domain is bounded the expression (5.19) is
integrable with respect to the measure in (LABEL:lim7). This allows one to
bring the $\zeta\rightarrow 0^{+}$ limit inside the triple integral. Taking
the $\zeta\rightarrow 0^{+}$ limit in (LABEL:lim7) only the $n=m=0$ term is
nonzero and results in

$\lim_{\zeta\rightarrow
0^{+}}\norm{\mathbb{F}_{\zeta}\psi}=\frac{1}{(4\pi)^{d+1}}\int\\!d^{d+1}y|g|^{3/2}(y)|\psi(y)|^{2}\bigg{|}\int\\!d^{d+1}\Delta
y\,e^{\frac{ie^{i\theta}}{4}g^{\theta}_{\mu\nu}(y)\Delta y^{\mu}\Delta
y^{\nu}}\bigg{|}^{2}$ (5.20)

The modulus-square of the Gaussian integral evaluates to
$(4\pi)^{d+1}/|g|(y)$, which yields (5.14). ∎

Global parametrix. Later on we seek to compare the parametrix $F_{\zeta}$ to
our globally defined kernel $K_{\zeta}$. To this end an extension of the local
parametrix (5.9) to all of $M\times M$ is needed; in particular, we want a
parametrix that does not just vanish along the diagonal outside a compact
region.

The extension is done using a partition of unity subordinate to a suitable
open cover of $M\times M$. We have already shown through Lemma 5.3 that every
point in $M$ has an open chart neighborhood on which a local parametrix with
properties (5.13), (5.14) can be defined. Since $M$ is assumed to have a
countable topological base, it can be covered by countably many such open
chart neighborhoods $(V_{l})_{l\in\mathbb{N}}$, where we may assume that no
$V_{l}$ is contained in any $V_{l^{\prime}}$ for $l\neq l$. In fact, we can
assume that each point in $M$ is contained in at most $d+2$ of the $V_{l}$’s.
This is because on a (smooth, second countable) manifold the Lebesgue covering
dimension coincides with the topological dimension. By construction, the
charts are such that the local parametrices $F^{(l)}_{\zeta}(y,y^{\prime})$,
$y,y^{\prime}\in V_{l}$, satisfy Lemma 5.3. Finally, since for fixed $\zeta>0$
the $F^{(l)}_{\zeta}$ are smooth we may also assume the neighborhoods to be
sufficiently small such that

$\displaystyle\int_{V_{l}}d\mu(y^{\prime})|F_{\zeta}^{(l)}(y,y^{\prime})|\leq\frac{C}{(4\pi\zeta)^{\frac{d+1}{2}}}\,,\quad\int_{V_{l}}d\mu(y)|F_{\zeta}^{(l)}(y,y^{\prime})|\leq\frac{C}{(4\pi\zeta)^{\frac{d+1}{2}}}\,,$
(5.21)

for all $y\in V_{l}$, with a $y$ and $l$ independent constant $C>0$.

The rectangular open neighborhoods $(V_{l}\times V_{l})_{l\in\mathbb{N}}$ then
cover the diagonal of $M\times M$. To get an open cover of $M\times M$ we
augment this by $W:=(M\times M)\backslash{\rm diag}(M\times M)$. Since ${\rm
diag}(M\times M)\subset M\times M$ is closed, $W$ is indeed open, and

$\\{W\\}\cup\\{V_{l}\times V_{l}\\}_{l\in\mathbb{N}}\,,$ (5.22)

furnishes an open cover of $M\times M$. Then, there exists a locally finite
partition of unity $(\chi_{l})_{l\in\mathbb{N}_{0}}$ that is subordinate to
this cover. Explicitly,

  * (i)

$\chi_{l}\in C^{\infty}(M\times M,[0,1])$, for all $l\in\mathbb{N}_{0}$.

  * (ii)

${\rm supp}\,\chi_{0}\subset W$, and ${\rm supp}\,\chi_{l}\subset V_{l}\times
V_{l}$, $l\in\mathbb{N}$.

  * (iii)

Every point in $M\times M$ has an open neighborhood that intersects only
finitely many of the $({\rm supp}\,\chi_{l})_{l\in\mathbb{N}_{0}}$. In
particular, ${\rm supp}\,\chi_{l}(y,\,\cdot\,)$ with $y\in M$ fixed is
nonempty for at most $d+2$ of the $l\geq 1$.

  * (iv)

$\sum_{l=0}^{\infty}\chi_{l}=1$ at every point in $M\times M$.

Using this partition of unity we define the global parametrix by

$F_{\zeta}(y,y^{\prime})=\sum_{l\geq
1}\chi_{l}(y,y^{\prime})F^{(l)}_{\zeta}(y,y^{\prime})\,,\quad y,y^{\prime}\in
M,\;\;\zeta>0\,,$ (5.23)

where the $l=0$ term does not appear, as the parametrix ought to be localized
around the diagonal. For simplicity we identify points $p,p^{\prime}\in V_{l}$
with their coordinates $y=\varphi_{l}(p)$,
$y^{\prime}=\varphi_{l}(p^{\prime})$, in the respective chart map
$\varphi_{l}$. By a slight abuse of notation we extend this to all
$p,p^{\prime}\in M$, with the relevant chart maps and their consistent
coordinatizations implicit. Note that for each $(y,y^{\prime})$ the sum over
$l$ is finite by property (iii) of the partition. For simplicity, we continue
to write $F_{\zeta}$ for the resulting bi-scalar parametrix on $M\times M$. It
is then jointly smooth in $\zeta,y,y^{\prime}$ and nonzero in a neighborhood
of the diagonal of $M\times M$. The diagonal $F_{\zeta}(y,y)$ is governed by
the glued diagonal coefficients

$A^{\theta}_{n}(y):=\sum_{l\geq
1}\chi_{l}(\varphi_{l}(p),\varphi_{l}(p))(A_{n}^{\theta})^{(l)}(\varphi_{l}(p),\varphi_{l}(p))\,,$
(5.24)

where on the right hand side we indicated the chart maps. By the smoothness
assumption these $A^{\theta}_{n}:M\rightarrow\mathbb{C}$ are globally defined.
The main result needed is the global counterpart of Lemma 5.3.

######  Proposition 5.4.

The global parametrix (5.23) has the property (a) from Lemma 5.3 but now for
$C_{c}^{\infty}(M)$ test functions. Likewise, property (b) from Lemma 5.3
holds, but now for all $\psi\in L^{2}(M)$.

Proof. (a) Let $\psi\in C_{c}^{\infty}(M)$. Then ${\rm supp}\,\psi$ intersects
the support of only finitely many $\chi_{l}(y,\,\cdot\,)\in
C^{\infty}(M,[0,1])$. To see this, consider $\\{y\\}\times{\rm
supp}\,\psi\subset M\times M$, which can be covered by open neighborhoods that
intersect only finitely many of the ${\rm supp}\,\chi_{l}$ (by property (iii)
of the partition). Since $\\{y\\}\times{\rm supp}\,\psi\subset M\times M$, is
compact, there exist a finite subcover of it by such neighborhoods. Hence
${\rm supp}\,\psi$ only intersects finitely many $\chi_{l}(y,\,\cdot\,)$, as
claimed, say $\chi_{l_{1}}(y,\,\cdot\,),\ldots,\chi_{l_{K}}(y,\,\cdot\,)$.

To proceed, let $\mathbb{F}_{\zeta}$ be the integral operator whose kernel is
the global parametrix (5.23). Its action on $\psi\in C_{c}^{\infty}(M)$ is
given by

$(\mathbb{F}_{\zeta}\psi)(y)=\sum_{k=1}^{K}\int_{V_{l_{k}}}\\!d\mu_{g}(y^{\prime})F_{\zeta}^{(l_{k})}(y,y^{\prime})\chi_{l_{k}}(y,y^{\prime})\psi(y^{\prime})\,,$
(5.25)

using ${\rm supp}\,\chi_{l_{k}}(y,\,\cdot\,)\subset V_{l_{k}}$. Note in
particular, that the image has compact support as well:
$\mathbb{F}_{\zeta}\psi\in C_{c}^{\infty}(M)$ for $\psi\in C_{c}^{\infty}(M)$.
Since each $\chi_{l}(y,\cdot)\in C_{c}^{\infty}(V_{l})$, Lemma 5.3(a) is
applicable to each term in the sum (5.25), giving

$\lim_{\zeta\rightarrow
0^{+}}(\mathbb{F}_{\zeta}\psi)(y)=\sum_{k=1}^{K}\chi_{l_{k}}(y,y)\psi(y)\,.$
(5.26)

By property (iv) of the partition of unity, the sum over all $\chi_{l}$ is
$1$, pointwise in $M\times M$. However, as seen above, among all $\chi_{l}$’s
only the $\chi_{l_{1}},\ldots,\chi_{l_{K}}$, are potentially supported at
$(y,y)$ with $y\in{\rm supp}\,\psi$. Further, $\chi_{0}(y,y)=0$ as
$(y,y)\in{\rm diag}(M\times M)$. Thus,
$1=\sum_{l=0}^{\infty}\chi_{l}(y,y)=\sum_{k=1}^{K}\chi_{l_{k}}(y,y)$, which
gives the assertion (a).

(b) We first use Holmgren’s criterion ([30], p.176) to show that (5.23)
defines a bounded integral operator $\mathbb{F}_{\zeta}:L^{2}(M)\rightarrow
L^{2}(M)$. According to the criterion, a sufficient condition for this to be
the case is that

$\sup_{y\in
M}\int\\!d\mu_{g}(y^{\prime})\,|F_{\zeta}(y,y^{\prime})|<\infty\quad\mbox{and}\quad\sup_{y^{\prime}\in
M}\int\\!d\mu_{g}(y)\,|F_{\zeta}(y,y^{\prime})|<\infty\,,$ (5.27)

where $F_{\zeta}$ is the global parametrix (5.23). We start with the first
condition and insert (5.23) to find

$\int\\!d\mu_{g}(y^{\prime})\,|F_{\zeta}(y,y^{\prime})|\leq\sum_{k=1}^{d+2}\int_{V_{l_{k}}}\\!d\mu_{g}(y^{\prime})\,\chi_{l_{k}}(y,y^{\prime})|F_{\zeta}^{(l_{k})}(y,y^{\prime})|\,.$
(5.28)

Initially, the sum extends over all $l\geq 1$. However, for any fixed $y$ the
second part of property (iii) ensures that at most $d+2$ open sets
$V_{l_{k}},k=1,\ldots,d+2$, contribute. By definition of the $\chi_{l}$,
$l\neq 0$, the support of $\chi_{l_{k}}(y,y^{\prime})$ is contained in
$V_{l_{k}}\times V_{l_{k}}$, even as $y^{\prime}$ varies. This gives (5.28).
Finally, using property (5.21) the right hand side of (5.28) can be bounded by
$C(d\\!+\\!2)(4\pi\zeta)^{-\frac{d+1}{2}}$, for a $y$ and $l_{k}$ independent
constant $C$. This gives the first bound in (5.27) for each fixed $\zeta>0$.
The flipped version is obtained similarly. This shows that
$\mathbb{F}_{\zeta}:L^{2}(M)\rightarrow L^{2}(M)$ is a bounded operator, for
all $\zeta>0$.

Next, consider the explicit form of $\|\mathbb{F}_{\zeta}\psi\|^{2}$,

$\displaystyle\int\\!d\mu_{g}(y)\,|(\mathbb{F}_{\zeta}\psi)(y)|^{2}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\int\\!d\mu_{g}(y)\,\int\\!d\mu_{g}(y_{1})\,\int\\!d\mu_{g}(y_{2})\,$
$\displaystyle\times$ $\displaystyle\sum_{l,m\geq
1}\chi_{l}(y,y_{1})\chi_{m}(y,y_{2})F^{(l)}_{\zeta}(y,y_{1})F^{(m)}_{\zeta}(y,y_{2})^{*}\psi(y_{1})\psi(y_{2})^{*}\,.$