# Functional formulation of quantum theory of a scalar field in a metric with
Lorentzian and Euclidean signatures

Z. Haba
Institute of Theoretical Physics, University of Wroclaw,
50-204 Wroclaw, Plac Maxa Borna 9, Poland
email:zbigniew.haba at uwr.edu.pl

###### Abstract

We study the Schrödinger equation in quantum field theory (QFT) in its
functional formulation. In this approach quantum correlation functions can be
expressed as classical expectation values over (complex) stochastic processes.
We obtain a stochastic representation of the Schrödinger time evolution on
Wentzel-Kramers-Brillouin (WKB) states by means of the Wiener integral. We
discuss QFT in a flat expanding metric and in de Sitter space-time. We
calculate the evolution kernel in an expanding flat metric in the real time
formulation. We discuss a field interaction in pseudoRiemannian and Riemannian
metrics showing that an inversion of the signature leads to some substantial
simplifications of the singularity problems in QFT.

##  1 Introduction

We discuss the functional integral approach to quantum field theory (QFT) in a
complete analogy to the Schrödinger picture in quantum mechanics [1, 2]. In
contradistinction to the Heisenberg picture, we insist on states and their
time evolution. The functional approach to QFT is a realization of the
conventional Schrödinger picture of quantum mechanics in the space of
functions of an infinite number of variables. Its efficiency has been
demonstrated in refs. [3, 4], in applications to quantum fields in de Sitter
space. In this paper, we discuss the functional approach to QFT on general
hyperbolic manifolds. In the functional formulation, we can develop the path
integral methods for a calculation of expectation values earlier exploited at
imaginary time as a rigorous tool in quantum mechanics [5]. We can extend
these methods to real time by means of complex-valued stochastic processes.
The extension to real time requires a complex extension of the configuration
(field) space. For fields on a manifold, such an extension means a
generalization of the path integral to complex manifolds (for some recent
attempts on such generalizations, see [6, 7, 8, 9, 10, 11, 12, 13, 14]). The
standard approach to field quantization on a manifold [15, 16] implicitly
assumes a non-unique ground state. The field can be expanded in
creation–annihilation operators. These operators are defined in the Fock space
which, in functional representation, is a Gaussian normalized state. The
functional representation of quantum fields does not require the Fock space.
We can define the Feynman path integral by means of the Wiener integral, which
provides a particular Gaussian functional measure for a field configuration
space (this could also be considered as a particular Fock space). Such a
formulation does not require that there is a Fock space as a ground state for
quantum fields. In this paper, the field is determined by a WKB Gaussian wave
function as a solution of the Schrödinger equation for the canonical
Hamiltonian defined by canonical quantization. We do not assume that the
particular WKB wave function is normalizable. Nevertheless, we can define
correlation functions in some other normalizable states with WKB phase
factors. These correlation functions determine the quantum fields. The
computation of correlations can be reduced to a calculation of expectation
values over stochastic processes. For this reason, the functional formulation
can be useful for numerical calculations. In earlier papers [17, 18], we have
derived a representation of the Feynman integral by means of the Brownian
motion in quantum mechanics, following an approach of mathematicians Cameron
and Doss [19, 20]. For quantum field theory, the paths must be transformed
into the paths of a quantum oscillator as a quantum field is a collection of
oscillators. For a free field in Minkowski space-time, we have discussed this
approach in [21], introducing a random field satisfying a stochastic equation
defined by the ground state. For a free field on a general Riemannian
manifold, a random field determined by a Gaussian state defined on the
Riemannian manifold is needed. It can be seen that, in order to obtain an
interacting field, a non-linear stochastic equation is required. The presence
of noise in semiclasical quantum gravity correlation functions has been
discovered in refs. [22, 23]. Our approach relies on the construction of a
Gaussian random field. Instead of looking for a non-linear stochastic equation
of the random field, we introduce an interaction via the Feynman–Kac formula.
There is no problem of such a construction on the perturbative level.
Perturbation theory reduced to a polynomial expansion in quantum fields will
be equivalent to the standard QFT (hence, the standard renormalization of
polynomials applies). However, there arises a difficult task to show that the
corresponding Feynman–Kac factor has a finite expectation value with respect
to the Wiener measure. Then, the functional integral over the solutions
defining the correlation functions must be finite. We show that a change in
the signature of the spatial metric (the time remains real) can substantially
facilitate the proof of integrability. The problem has been studied in
constructive quantum field theory in the Euclidean framework. In the Feynman
integral formulation (real time), we deal with oscillatory integrals. In such
a case, some stability problems can be avoided. We assume that time is well-
defined in an evolution of the quantum scalar field on a classical
gravitational background. The appearance of time and the Schrödinger evolution
in a semiclassical framework for quantum gravity have been discussed in [24,
25, 26, 27, 28]. We repeat (in a somewhat modified way) a derivation of the
Schrödinger equation from the Wheeler–DeWitt equation [29] in Appendix H. We
hope that the study of the Schrödinger equation in an external metric can help
to solve the Wheeler–DeWitt constraint in quantum gravity.

The plan of this paper is the following. First, we explain the mathematically
rigorous probabilistic method [30, 31] of solving the imaginary time
Schrödinger equation for a quantum (Euclidean) scalar field as a perturbation
of the ground state solution . Then, in Section 3, the method is extended by
an analytic continuation to apply to solutions for the real-time Schrödinger
equation. We discuss the functional Schrödinger equation for a scalar field on
a globally hyperbolic manifold [32, 33]. We consider general complex metrics
which may arise as saddle points in an average over metrics in quantum
gravity. In Section 4, we show that a free scalar quantum field has the
Schrödinger wave function solution, which is a pure WKB Gaussian phase if the
initial state is a pure Gaussian phase. Using this Gaussian solution, we
construct a random wave field (as a solution of a stochastic equation), which
allows for the construction of a general solution of the Schrödinger equation
as an expectation value over the Brownian motion. The solution can be
calculated in a simple way if the initial state is of the form of the Gaussian
factor times a mild perturbation. It can have an explicit form if the manifold
has a large group of symmetry. In Section 5, we discuss the scalar field on de
Sitter space-time and on its Euclidean continuation (the sphere $S^{4}$).
Stochastic equations for the de Sitter field and fields in an expanding
(homogeneous) universe are discussed in Section 6. The large symmetry group
allows for an expansion in terms of eigenfunctions of the algebra of this
group. In this section, some field correlations are calculated. The explicit
Gaussian solutions of the Schrödinger equation are not available for general
initial conditions. In Section 6, we derive an asymptotic formula for the
solution at a large angular momentum. An analogous formula at large momenta
for the homogeneous expanding metric is obtained in Appendix B. The asymptotic
formulas allow for an approximate calculation of correlation functions of
stochastic fields. In Section 7, the free field time evolution is expressed by
an evolution kernel. Then, a computation of multi-time correlation functions
can be reduced to Gaussian integrals with the evolution kernels. The method
can be extended to interacting fields by means of the Feynman–Kac formula. The
correlation functions determine QFT (Wightman’s reconstruction). In Sections
8–10, the power-law evolution of massless scalar fields is discussed. The most
detailed results concern the free field in a radiation background (Section 10)
with the spatial metric $g_{jk}\simeq\delta_{jk}t$, which changes signature at
$t=0$. In Sections 11 and 12, we discuss polynomial and trigonometric
interactions. We show, in Section 11, that for a finite mode approximation the
expectation value of the Feynman–Kac factor can be finite in the Wiener
integral formulation of the Feynman integral when we average the potentials
over the random fields constructed from the Gaussian WKB wave functions in the
earlier sections. In Section 12, it is shown that an inversion of the spatial
signature allows to show that the expectation value of the Feynman–Kac factor
is finite for an infinite number of modes. The inversion of the signature may
be considered either as a technical tool or as a quantum effect of an average
of the functional integral over quantum metrics. Some supplementary material
is located in the Appendices A-H. Appendices C-E describe for the convenience
of the reader some simplified models (discussed earlier) of the ones studied
in the main part. Appendices A, B, F and G contain some additions to the
results in the main text, which illustrate the method of stochastic equations
(this is just a transformation of sample paths). We have studied the Brownian
motion formulation of quantum physics for some time. The motivation was
twofold: 1. to formulate the Feynman integral as a rigorous mathematical tool,
and 2. to make quantum correlations susceptible to standard simulations by
means of random variables. We obtained the stochastic formulation of the
Feynman integral first in quantum mechanics [17, 18, 34]. The stochastic free
field is introduced in [21, 35, 36]. The inversion of the metric in a
radiation background is discussed in [37]. In this paper, we present a
comprehensive approach to the stochastic representation of QFT (beyond the
homogeneous expanding metric), including the previously studied models as
special cases. The choice of the stochastic field is not unique. It depends on
the selection of the Gaussian WKB state. We discuss, in detail, the form of
the field which enables a construction of a well-defined (integrable)
Feynman–Kac factor for the field theory with an interaction.

##  2 The Imaginary Time Schrödinger Equation

First, we discuss the standard canonical field theory in a mathematically
rigorous imaginary time formulation [30]. The Hamiltonian is defined as

${\cal H}=\frac{1}{2}\int d{\bf x}\Big{(}\Pi^{2}+(\nu\Phi)^{2}\Big{)}+\int
d{\bf x}V(\Phi)={\cal H}_{0}+V,$ (1)

where

$\nu=\sqrt{-\triangle+M^{2}}.$ (2)

The canonical momentum $\Pi$ satisfies the commutation relations with the
field $\Phi$

$[\Phi({\bf x}),\Pi({\bf y})]=i\hbar\delta({\bf x}-{\bf y}).$ (3)

Let $\psi_{t}^{g}$ be a solution (usually the ground state) of the imaginary
time Schrödinger equation

$-\hbar\partial_{t}\psi={\cal H}\psi.$ (4)

Let us consider the general solution of the Schrödinger Equation (4) with the
initial condition $\psi=\psi_{0}^{g}\chi$

$\psi_{t}=\psi_{t}^{g}\chi_{t}.$ (5)

Then, $\chi_{t}$ satisfies the equation

$-\hbar\partial_{t}\chi=\int d{\bf
x}\Big{(}\frac{1}{2}\Pi^{2}-(\Pi\ln\psi_{t}^{g})\Pi\Big{)}\chi$ (6)

with

$\Pi({\bf x})=-i\hbar\frac{\delta}{\delta\Phi({\bf x})}.$ (7)

$\psi_{t}^{g}$ may be an arbitrary solution of the Schrödinger Equation (4)
(with an arbitrary initial condition). The efficiency of the representation
((5) and (6)) depends on the choice of $\psi_{t}^{g}$ and the assumption that
initial states are under consideration (these will be the WKB states). It can
be seen that, in Equation (6), the kinetic term $(\nu\Phi)^{2}$, as well as
the potential $V$, are absent. Equation (6) is a diffusion equation in
infinite dimensional spaces [31]. The solution of Equation (6) can be
expressed as [38]

$\chi_{t}(\Phi)=E\Big{[}\chi\Big{(}\Phi_{t}(\Phi)\Big{)}\Big{]},$ (8)

where $\Phi_{t}(\Phi)$ is the solution of the stochastic equation ($t\geq
s\geq 0$)

$d\Phi_{s}({\bf x})=\hbar\frac{\delta}{\delta\Phi_{s}({\bf
x})}\ln\psi_{t-s}^{g}dt+\sqrt{\hbar}dW_{s}({\bf x})$ (9)

with the initial condition $\Phi$. $E[\ldots]$ denotes an expectation value
with respect to the Wiener process (Brownian motion) with the mean zero and
the covariance ($t\geq 0$, $s\geq 0$)

$E\Big{[}W_{t}({\bf x})W_{s}({\bf y})\Big{]}=min(t,s)\delta({\bf x}-{\bf y}).$
(10)

The correlation functions of the quantum Euclidean field in the state
$\psi_{0}^{g}$ can be expressed by the correlation functions of the stochastic
process $\Phi_{t}$. If we could find a particular solution of the Schrödinger
equation and solve the non-linear stochastic differential Equation (9), then
the problem of solving QFT and calculating the field correlation functions
could be reduced to a calculation of expectation values with respect to the
Wiener process. We do not know any solution of the Schrödinger equation for
scalar field theory (for a potential $V(\Phi)$ which is not quadratic).
However, in field theories with large symmetry, this could be possible (let us
mention Chern–Simons states in gauge theories [6] and Kodama states in gravity
[39]). The imaginary time in this section has a rigorous mathematical
formulation for super-renormalizable field theories in two dimensions [30]. In
general, for arbitrary states $\psi_{t}^{g}$, higher dimensions and real time,
we expect difficulties with a derivation of solutions of Equation (9) and
their renormalization. We can manage in this paper linear stochastic equations
(corresponding to Gaussian $\psi_{t}^{g}$) for free field theory. Then, the
interaction is introduced as usual by means of the Feynman–Kac formula.

Let us consider the simplest example: the free field. Then, the ground state
is

$\psi^{g}=Z^{-1}\exp(-\frac{1}{2\hbar}\Phi\nu\Phi),$ (11)

where $Z$ is the state normalization. The stochastic Equation (9) reads

$d\Phi_{t}=-\nu\Phi_{t}dt+\sqrt{\hbar}dW_{t}.$ (12)

The solution is (with the initial condition $\Phi$ at $t_{0}$)

$\Phi_{t}=\exp(-\nu(t-t_{0}))\Phi+\sqrt{\hbar}\int_{t_{0}}^{t}\exp(-\nu(t-s))dW_{s}.$
(13)

We calculate

$\begin{array}[]{l}\int d\Phi\psi_{g}^{2}E\Big{[}\exp(\int dtd{\bf
x}f_{t}({\bf x})\Phi_{t}({\bf x}))\Big{]}\cr=\exp\Big{(}\frac{1}{2}\int
dtdt^{\prime}\Big{(}f_{t},(2\nu)^{-1}\exp(-\nu|t-t^{\prime}|)f_{t^{\prime}}\Big{)}\Big{)}\end{array}$
(14)

On the rhs of Equation (14), we have the generating functional for the
correlation functions of the quantum Euclidean free field.

As a time-dependent solution of Equation (4) (with $V=0$), we may consider

$\psi_{t}^{g}=A(t)\exp(i\frac{1}{2\hbar}(\Phi\Gamma_{t}\Phi)).$ (15)

In Equation (15), $\Gamma$ is an operator with an integral kernel $\Gamma({\bf
x},{\bf y})$. We can derive an equation for this operator, demanding that (15)
is the solution of the Schrödinger Equation (4).

Then, $\psi_{t}^{g}$ is the solution of the free imaginary time Schrödinger
Equation (4) ($V=0$) if

$i\partial_{t}\Gamma+\Gamma^{2}+\nu^{2}=0.$ (16)

Equation (16) is equivalent to

$(\partial_{t}^{2}-\nu^{2})u=0$ (17)

if

$-i\Gamma=u^{-1}\frac{d}{dt}u.$ (18)

The general solution of Equation (17) is

$u=C_{1}\sinh(\nu t)+C_{2}\cosh(\nu t)$ (19)

If $C_{1}=C_{2}$, then, from Equations (15),(18) and (19) we obtain the ground
state solution (11). We discuss the case $C_{1}=0$ in Appendix A. It defines
another field $\Phi_{t}$ whose correlation functions are equal to the ones of
the standard free Euclidean field, but in another time-dependent state (15).
We obtain another realization of the solution (5) of the Schrödinger equation
for the free field.

With the potential $V$ in Equation (1), the solution of the Schrödinger
Equation (4) reads [5, 38] (the Feynman–Kac formula requires $V(\Phi)$ to be
bounded from below)

$\chi_{t}(\Phi)=E\Big{[}\exp\Big{(}-\frac{1}{\hbar}\int_{0}^{t}V(\Phi_{s})ds\Big{)}\chi(\Phi_{t}(\Phi))\Big{]}.$
(20)

At the end of this section, we wish to point out some problems with the
definition of the Feynman–Kac integral (20) in Euclidean field theory (even if
$V(\Phi)$ is bounded from below). We consider in subsequent sections
exponential potentials

$V(\Phi)=\lambda\int d{\bf x}\exp(\alpha\Phi({\bf x})),$

where $B\subset R^{d}$ is a bounded region in $R^{d}$. It can be seen that
higher orders of the perturbation expansion in $\lambda$ of the normal ordered
exponentials in the Feynman–Kac Formula (20) are divergent if the dimension of
the space-time is $d\geq 3$, e.g., in the second order ($:-:$ denotes the
normal ordering), we obtain

$\begin{array}[]{l}\lambda^{2}\int_{0}^{t}ds\int_{0}^{t}d\tau\int_{B}d{\bf
x}\int_{B}d{\bf y}E\Big{[}:\exp(\alpha\Phi_{s}({\bf
x}))::\exp(\alpha\Phi_{\tau}({\bf
y})):\Big{]}\cr=\lambda^{2}\int_{0}^{t}ds\int_{0}^{t}d\tau\int_{B}d{\bf
x}\int_{B}d{\bf y}\exp\Big{(}\alpha^{2}E\Big{[}\Phi_{s}({\bf
x})\Phi_{\tau}({\bf y})\Big{]}\Big{)}.\end{array}$

The two-point function is positive and at short distances

$E\Big{[}\Phi_{s}({\bf x})\Phi_{\tau}({\bf
y})\Big{]}\simeq\Big{(}\Big{(}(s-\tau)^{2}+({\bf x}-{\bf
y})^{2}\Big{)}^{1-\frac{d}{2}}.$

The $\lambda^{2}$ term of the perturbation series of Equation (20) is infinite
if $\alpha$ is real and $d\geq 3$ (this follows from $\exp(x)\geq
1+\frac{x^{4}}{4!}$ for $x\geq 0$).

If $\alpha=i\beta$ is purely imaginary, then

$\begin{array}[]{l}\lambda^{2}t^{2}|B|^{2}\geq\lambda^{2}\int_{0}^{t}ds\int_{0}^{t}d\tau\int_{B}d{\bf
x}\int_{B}d{\bf y}E\Big{[}:\exp(i\beta\Phi_{s}({\bf
x}))::\exp(i\beta\Phi_{\tau}({\bf
y})):\Big{]}\cr=\lambda^{2}\int_{0}^{t}ds\int_{0}^{t}d\tau\int_{B}d{\bf
x}\int_{B}d{\bf y}\exp\Big{(}-\beta^{2}E\Big{[}\Phi_{s}({\bf
x})\Phi_{\tau}({\bf
y})\Big{]}\Big{)}\cr\geq\lambda^{2}t^{2}|B|^{2}\exp\Big{(}-\beta^{2}t^{-2}|B|^{-2}\int_{0}^{t}ds\int_{0}^{t}d\tau\int_{B}d{\bf
x}\int_{B}d{\bf y}E\Big{[}\Phi_{s}({\bf x})\Phi_{\tau}({\bf
y})\Big{]}\Big{)}\end{array}$

from Jensen inequality. We obtain an upper bound and a non-zero lower bound if
$d<6$. However, if we form real potentials as, e.g., $\lambda\cos\beta\Phi$,
then at the order $\lambda^{2}$, there will be terms without an upper bound
because of the multiplication of the terms $\exp(i\beta\Phi({\bf x}))$ with
$\exp(-i\beta\Phi({\bf y}))$.

##  3 Real Time: The Free Field on a Manifold

We consider a globally hyperbolic manifold [33] with a choice of coordinates
such that the metric is of the form

$ds^{2}=g_{00}dx^{0}dx^{0}-g_{jk}dx^{j}dx^{k}.$ (21)

The Lagrangian of the free field is [40]

$L=\frac{1}{2}\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{M^{2}}{2}\sqrt{-g}\phi^{2},$
(22)

where $g=\det(g_{\mu\nu})$. The canonical momentum is

$\Pi=g^{00}\sqrt{-g}\partial_{0}\phi.$ (23)

From the Lagrangian (22), we drive the canonical Hamiltonian

$\begin{array}[]{l}{\cal H}(t)\equiv\int d{\bf x}{\cal H}(g,{\bf x})=\int
d{\bf x}\Big{(}\Pi\partial_{0}\phi-L\Big{)}\cr=\int d{\bf
x}\Big{(}\frac{1}{2}g_{00}\frac{1}{\sqrt{-g}}\Pi^{2}+\frac{1}{2}\sqrt{-g}g^{jk}\partial_{j}\phi\partial_{k}\phi+\frac{M^{2}}{2}\sqrt{-g}\phi^{2}\Big{)}.\end{array}$
(24)

In subsequent sections, we shall discuss the Lagrangian (22) and the
Hamiltonian (24) for the metric tensors $g_{\mu\nu}$, which can arise as
saddle points in the Feynman path integral. Such metrics satisfy Einstein
equations, but they do not fulfill the requirement $-g>0$. We must choose the
square root $\sqrt{-g}$ in such a way (see [6, 7, 9, 11, 10, 12]) that the
path integral and the Schrödinger equation

$i\hbar\partial_{t}\psi_{t}={\cal H}(t)\psi_{t}$ (25)

are well-defined. The solution of Equation (25) must define an operator which
is a contraction in a Hilbert space (otherwise $\psi_{t}$ may have an infinite
norm). We assume (as in [7]) that, for $t\geq 0$, the metric is real and
Lorentzian. In the past, the stationary points of the Lagrangian with matter
satisfying some positive energy conditions [32] will necessarily exhibit a Big
Bang singularity (then, e.g., $\frac{1}{\sqrt{-g}}$ is not defined). We assume
that the solutions of Einstein equations have a continuation to $t<0$, but do
not satisfy the requirements of the classical general relativity (they may be
Euclidean or even complex). So, for positive time, we shall have the
Schrödinger equation, whereas for negative time with the Hamiltonian (24), a
diffusion-type Equation (a contraction [41]). The necessary condition for a
contraction at $t<0$ is that the infinite dimensional diffusion generator has
the imaginary part of the second order differential operator, which is a
positive operator. From the Hamiltonian (24), we can see that this will be the
case if

$g_{00}\frac{1}{\sqrt{-g}}=R+iI$ where $I\geq 0$,

where $R$ and $I$ are the real and imaginary parts of a complex function. In
such a case, we write ${\cal H}=i{\cal H}_{E}$ and write the Schrödinger
Equation (25) as a diffusion equation

$\hbar\partial_{t}\psi_{t}={\cal H}_{E}(t)\psi_{t}$ (26)

In our models, we choose $\sqrt{-g}=-i\sqrt{|g|}$. Then, ${\cal H}_{E}$ will
be (on a formal level) a Hermitian positive operator. Hence, Equation (26)
defines a contraction (diffusion) for $t<0$. For the inverted metric, the mass
term has an opposite sign to the kinetic term. So, we invert its sign as
$M^{2}\rightarrow-\mu^{2}$.

In Section 10, we discuss a solution for a homogeneous radiation metric
$a^{2}\simeq t$ which leads to an infinite energy density when $t\rightarrow
0$, but it is also a solution of Einstein equations at $t<0$ with $a^{2}<0$.
Examples which have a continuation from the real time in de Sitter space to
the four-sphere at the imaginary time have been discussed in [7, 8, 14].

Let us still mention another interpretation of the Hamiltonian (24) for an
inverted metric

$\begin{array}[]{l}{\cal H}(t)=\int d{\bf
x}\Big{(}\Pi\partial_{0}\phi-L\Big{)}\cr=\int d{\bf
x}\Big{(}\frac{1}{2}g_{00}\frac{1}{\sqrt{|g|}}\Pi^{2}+\frac{1}{2}\sqrt{|g|}g^{jk}\partial_{j}\phi\partial_{k}\phi+\frac{M^{2}}{2}\sqrt{|g|}\phi^{2}\Big{)}.\end{array}$
(27)

Such a modification of the Hamiltonian (24) transforms the Schrödinger
evolution with the Lorentzian metric into the one with an Euclidean metric. It
is still unitary. The inversion of the spatial metric in Equation (27) (so
that $g^{jk}$ is negatively definite) at $t=0$ is an analog of a
transformation of the oscillator for $t\geq 0$ into an upside-down oscillator
for $t<0$ [42, 43, 36]. The Hamiltonian (27) results from the replacement
$\sqrt{-g}\rightarrow\sqrt{|g|}$ in the Lagrangian (22) as a possible
candidate for quantum gravity.

Inserting the WKB wave function (15) in Equation (25), we obtain an equation
for the operator $\Gamma$ and the normalization coefficient $A$

$\partial_{t}\Gamma_{t}+\Gamma_{t}{\cal J}\Gamma_{t}+{\cal M}=0,$ (28)
$\partial_{t}\ln A=-\frac{1}{2}\int d{\bf
x}\frac{g_{00}}{\sqrt{-g}}\Gamma_{t}({\bf x},{\bf x}),$ (29)

where

${\cal J}({\bf y},{\bf y}^{\prime})=\frac{g_{00}}{\sqrt{-g}}\delta({\bf
y},{\bf y}^{\prime})$ (30)

and ${\cal M}$ is the differential operator

${\cal M}={\cal
J}^{-1}K^{2}=M^{2}\sqrt{-g}-\partial_{j}\sqrt{-g}g^{jk}\partial_{k}.$ (31)

where the operator [40]

$K^{2}=-g_{00}\frac{1}{\sqrt{-g}}\partial_{j}g^{jk}\sqrt{-g}\partial_{k}+g_{00}M^{2}$

is self-adjoint in $L^{2}(d\mu)$ with respect to the measure

$d\mu=g^{00}\sqrt{-g}d{\bf x}.$

Let us define the operator

${\cal G}_{t}=\exp(\int^{t}{\cal J}_{s}\Gamma_{s}ds).$ (32)

$\Gamma_{t}$ can be expressed by ${\cal G}_{t}$

$\Gamma_{t}={\cal J}_{t}^{-1}\partial_{t}{\cal G}{\cal G}^{-1}={\cal
J}_{t}^{-1}\partial_{t}\ln{\cal G}.$ (33)

${\cal G}_{t}$ satisfies a linear operator equation

$\partial_{t}^{2}{\cal G}=\partial_{t}{\cal J}{\cal J}^{-1}\partial_{t}{\cal
G}-{\cal J}{\cal M}{\cal G}.$ (34)

Let us note that $\sqrt{-g}$ cancels in ${\cal J}{\cal M}$ in Equation (34).
Hence, this equation does not depend on the interpretation of $\sqrt{-g}$ for
an inverted metric. It can be shown that the operator Equation (34) coincides
with the wave equation corresponding to the Lagrangian (22). Then, as
discussed in Section 2, $\Gamma\Phi$ is the drift in the stochastic Equation
(9). For the general metric (21), it is not simple to find a solution
$\Gamma$. In subsequent sections, we find explicit solutions assuming the
invariance either with respect to the de Sitter group or under translations
(homogeneous spatially flat manifold). Some information about $\Gamma$ is
needed in order to determine whether $\psi_{t}^{g}$ is square integrable. We
need to know $\Gamma$ for a construction of an interaction in such a way that
the Feynman–Kac factor has a finite expectation value.

We express the Feynman path integral solution of the Schrödinger Equation (25)
($V=0$) with the initial condition

$\psi^{g}_{0}=\exp(\frac{i}{\hbar}S_{0})$ (35)

in the form

$\begin{array}[]{l}\psi_{t}^{g}(\Phi)=\int
d\Phi(.)\exp\Big{(}\frac{i}{2\hbar}\int d\tau d{\bf
x}\Big{(}\sqrt{-g}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-M^{2}\sqrt{-g}\Phi^{2}\Big{)}\exp(\frac{i}{\hbar}S_{0}(\Phi_{t}(\Phi)).\end{array}$
(36)

Formula (36) is well-established with real values of $\sqrt{-g}$ as a solution
of Equation (25) [38]. However, a formal derivation of Equation (36) does not
use any assumption on the signature of the metric, as long as the exponential
(36) is bounded as a function of $\Phi$. As in the Hamiltonian Equation (25),
we assume that, for $t\geq 0$, the metric is real and Lorentzian. For $t<0$,
we admit complex $g_{\mu\nu}$ and complex $\sqrt{-g}$. If we require that the
quadratic factor $(\partial_{0}\Phi)^{2}$ in the Feynman integral (36) does
not grow for $t<0$, then the conditions upon the metric will be

$\sqrt{-g}g^{00}=\tilde{R}+i\tilde{I}$ where $\tilde{I}\leq 0$.

If the spatial part in the action in the exponential (36) is to be positive
definite for $t<0$, then

$\sqrt{-g}g^{jl}=R^{jl}+iI^{jl}$,

where $I^{jl}$ is a positive definite matrix. If $\sqrt{-g}=-i\sqrt{|g|}$,
then the real part of the integral of the quadratic term
$(\partial_{0}\Phi)^{2}$ in Equation (36) is negative, and if $g^{jl}$ is
negatively definite (inverted metric), then the real part of the spatial
quadratic term in (36) is also negative. The mass term has an opposite sign;
hence, if it is to be negative, we must change $M^{2}\rightarrow-\mu^{2}$ for
$t<0$. The conditions for the path integral coincide with the ones derived
from the Hamiltonian (below Equation (25)). We shall still discuss the
Schrödinger equations and path integrals in detail in specific models in
subsequent sections. The requirements for the complex metric have been
discussed in [12, 6, 13]. They look different then the ones required for the
Hamiltonian at the beginning of this section. We shall discuss these
conditions in Sections 9 and 10 when we discuss a change in signature.

We assume that $S_{0}$ is a quadratic form in $\Phi$

$S_{0}(\Phi)=(\Phi,\Gamma_{0}\Phi),$ (37)

where $(,)$ denotes the scalar product in $L^{2}(d{\bf x})$. We solve the
Schrödinger Equation (25) by means of the stationary phase method. We expand
the Feynman integral (36) around the stationary point $\phi_{s}^{c}(\Phi)$.
The stationary point is obtained as a solution $\phi_{s}^{c}(\Phi)$ of the
Cauchy problem with the initial field value $\Phi$, and the final boundary
condition on the time derivative [44]

$\frac{d\phi_{t}^{c}}{dt}=-\frac{\delta
S_{0}(\phi)}{\delta\phi}(\phi_{t}^{c}).$ (38)

The solution $\phi_{t}^{c}(\Phi)$ is linear in $\Phi$. We write

$\Phi_{s}=\phi_{s}^{c}(\Phi)+\sqrt{\hbar}\phi^{q}_{s}.$ (39)

Then,

$\psi_{t}^{g}(\Phi)=A_{t}\exp(\frac{i}{\hbar}S_{t}(\phi_{c}(\Phi))\equiv
A_{t}\exp(\frac{i}{2\hbar}(\Phi,\Gamma_{t}\Phi)).$ (40)

The classical action $S_{t}$ is a bilinear form in $\Phi$ defined by a real
kernel $\Gamma_{t}({\bf x},{\bf y})$ (if $S_{0}$ is a real function), $A_{t}$
is expressed by the determinant of an operator defined by the quadratic form
in $\phi^{q}$. The determinant depends only on time and the metric. It follows
that if $S_{0}$ is real, then $S_{t}$ is a real bilinear form (if $S_{0}$ is
complex, then $S_{t}$ is also complex). Hence, $\psi_{t}^{g}$, as a function
of $\Phi$, is a pure phase factor.

##  4 Gaussian Solution of the Schrödinger Equation and Linear Stochastic
Equations

We approach the quantum field theory (QFT) in Minkowski space-time by means of
stochastic equations [35, 21, 34]. The stochastic equations determine the
solution of the Schrödinger equation as in Equation (8). We continue the
imaginary time in Equations (8) and (9) to the real time [21]

$d\Phi_{s}({\bf x})=i\hbar\frac{\delta}{\delta\Phi_{s}({\bf
x})}\ln\psi_{t-s}^{g}dt+\sqrt{i\hbar}dW_{s}({\bf x}),$ (41)

where $\sqrt{i}=\exp(i\frac{\pi}{4})=\frac{1}{\sqrt{2}}(1+i)$. Let
$\hat{\Phi}$ be the relativistic quantum free field. Then, the generating
functional of the time-ordered correlation functions of $\hat{\Phi}$ in the
vacuum $\psi^{g}$ (11) can be expressed [21] by the solution $\Phi_{t}(\Phi)$
of the stochastic Equation (41)

$\begin{array}[]{l}(\psi_{g},T\Big{(}\exp\int dtd{\bf x}\hat{\Phi}_{t}({\bf
x})f_{t}({\bf x})\Big{)}\psi_{g})=\int
d\Phi\psi_{g}^{2}(\Phi)E\Big{[}\exp(\int dtd{\bf x}f_{t}({\bf
x})\Phi_{t}(\Phi,{\bf x}))\Big{]}\cr=\exp\Big{(}\frac{1}{2}\int
dtdt^{\prime}\Big{(}f_{t},(2\omega)^{-1}\exp(-i\omega|t-t^{\prime}|)f_{t^{\prime}}\Big{)}\Big{)},\end{array}$
(42)

where $T(\ldots)$ is the time-ordered product and $(f,g)$ denotes the scalar
product in $L^{2}(d{\bf x})$ .

When the initial condition is $\psi=\psi_{0}^{g}\chi$, and $\psi_{t}^{g}$ is
the solution of the Schrödinger Equation (25), then $\chi$ solves the equation

$i\hbar\partial_{t}\chi_{t}=\int d{\bf
x}\Big{(}\frac{1}{2}\Pi^{2}+(\Pi\ln\psi_{t}^{g})\Pi\Big{)}\chi_{t}.$ (43)

If the solution $\psi_{t}^{g}$ of the Schrödinger equation for quantum fields
defined on the Minkowski space-time is of the form (40) then Equation (41)
reads [38]

$d\Phi_{s}=-\Gamma_{t-s}\Phi_{s}ds+\sqrt{i\hbar}dW_{s}.$ (44)

Let $\Phi_{s}(\Phi)$ be the solution of Equation (44) with the initial
condition $\Phi$. If $\chi$ is a holomorphic function, then the solution of
the Schrödinger Equation (25) is

$\psi_{t}=\psi^{g}_{t}E\Big{[}\chi\Big{(}\Phi_{t}(\Phi)\Big{)}\Big{]}.$ (45)

With the interaction $V_{t}$, the Feynman formula reads

$\psi_{t}=\psi^{g}_{t}E\Big{[}\exp\Big{(}-\frac{i}{\hbar}\int_{0}^{t}V_{t-s}(\Phi_{s})ds\Big{)}\chi\Big{(}\Phi_{t}(\Phi)\Big{)}\Big{]}.$
(46)

The solution of the stochastic equation determines the free field correlation
functions ($V=0$)

$\begin{array}[]{l}(\psi_{0}^{g},F_{1}(\Phi_{t})F_{2}(\Phi)\psi_{0}^{g})=\int
d\Phi|\psi_{t}^{g}(\Phi)|^{2}F_{1}(\Phi)E\Big{[}F_{2}\Big{(}\Phi_{t}(\Phi)\Big{)}\Big{]}.\end{array}$
(47)

For more general states $\psi^{g}\chi$ and $V\neq 0$, we have

$\begin{array}[]{l}(\psi_{0}^{g}\chi,F_{1}(\Phi_{t})F_{2}(\Phi)\psi_{0}^{g}\chi)=\int
d\Phi|\psi_{t}^{g}(\Phi)|^{2}F_{1}(\Phi)\cr
E\Big{[}\chi\Big{(}\Phi_{t}(\Phi)\Big{)}\exp(-\frac{i}{\hbar}\int_{0}^{t}V_{t-s}(\Phi_{s})ds)\Big{]}^{*}\cr\times
E\Big{[}F_{2}\Big{(}\Phi_{t}(\Phi)\Big{)}\chi\Big{(}\Phi_{t}(\Phi)\Big{)}\exp(-\frac{i}{\hbar}\int_{0}^{t}V_{t-s}(\Phi_{s})ds)\Big{]}.\end{array}$
(48)

We generalize these correlation functions in Section 7 to multitime
correlation functions after a derivation of a formula for the evolution
propagator. In principle, QFT (with the Hilbert space and quantum fields) can
be determined by the correlation functions (Wightman construction).

##  5 De Sitter Space

In this and subsequent sections, we obtain quantum field evolution on the
Lorentzian and Euclidean backgrounds. First, we discuss particular solutions
(de Sitter and the sphere $S^{4}$) of Einstein equations with a cosmological
constant (obtained in [7, 8]). De Sitter space-time can describe an early
inflationary stage of the universe, as well as the final stage of an
acceleration driven by dark energy. The authors [7, 8] glue together the
Lorentzian solution for positive time with the Euclidean solution ($S^{4}$)
for an imaginary time. We consider several coordinate systems on de Sitter
space [45, 7] which can be considered as a submanifold of the complex quadric

$z_{1}^{2}+z_{2}^{2}+z_{3}^{3}+z_{4}^{2}+z_{5}^{2}=\frac{1}{H^{2}},$ (49)

where $H$ has the meaning of the Hubble constant. We first consider a real
form of the quadric (49), defining the de Sitter space

$x_{1}^{2}+x_{2}^{2}+x_{3}^{3}+x_{4}^{2}-x_{5}^{2}=\frac{1}{H^{2}}.$ (50)

In the coordinates $(t,\omega)$, where $\omega\in S^{3}$, the metric on the
hyper-sphere (50) is

$ds^{2}=dt^{2}-\frac{1}{H^{2}}\cosh^{2}(Ht)d\omega^{2},$ (51)

where $d\omega^{2}$ is the metric on $S^{3}$.

In conformal coordinates

$\cos(\tau)=\frac{1}{\cosh(Ht)},$ (52)

where $0<\tau<\frac{\pi}{2}$, we obtain the metric

$ds^{2}=\frac{1}{H^{2}\cos^{2}(\tau)}(d\tau^{2}-d\omega^{2}).$ (53)

The Euclidean version of the manifold (53) describes the four-dimensional
sphere of radius $\frac{1}{H}$

$x_{1}^{2}+x_{2}^{2}+x_{3}^{3}+x_{4}^{2}+x_{5}^{2}=\frac{1}{H^{2}}$

Then, the metric is

$ds^{2}=dt^{2}+\frac{1}{H^{2}}\cos^{2}(Ht)d\omega^{2}.$ (54)

The introduction of conformal coordinates

$\cosh(\tau)=\frac{1}{\cos(Ht)},$ (55)

where $-\frac{\pi}{2H}<t\leq 0$ (we choose negative time in the Euclidean
domain) gives the metric

$ds^{2}=\frac{1}{H^{2}\cosh^{2}(\tau)}(d\tau^{2}+d\omega^{2}).$ (56)

We can also introduce spatially flat coordinates describing the expanding
universe (which will be discussed in detail in subsequent sections)

$ds^{2}=dt^{2}-a^{2}d{\bf x}^{2}.$

The expanding flat metric in coordinates which cover the half of de Sitter
space-time (visible by an observer at the origin) is

$ds^{2}=dt^{2}-\exp(2Ht)d{\bf x}^{2}.$ (57)

The “Euclidean” version of the metric (57) (${\bf x}\rightarrow i{\bf x}$)

$ds^{2}=dt^{2}+\exp(2Ht)d{\bf x}^{2}$ (58)

does not represent a metric on the sphere $S^{4}$. This is the metric on the
hyperbolic space which is the Euclidean version of the anti-de Sitter space-
time. In fact, the metric (58) can be expressed in a form familiar from the
realization of the anti-de Sitter space as a generalized Poincare upper half-
plane

$ds^{2}=y^{-2}(H^{-2}dy^{2}+d{\bf x}^{2})$

with $y=\exp(-Ht)$. The sphere $S^{4}$, Euclidean anti-de Sitter space and
Euclidean continuations of de Sitter space are closely related [46]. We are
allowed to treat the coordinates $t$ and $\tau$ in Equations (51)–(58) in
Lorentzian and Euclidean metrics as time in the Lagrangian formalism (in the
Euclidean version, the time evolution will be a rotation of the sphere).

There remains to study the Schrödinger Equations (25)–(27) resulting from the
definition of ${\cal H}$ in Equations (24) and (27). First, we look for a
Gaussian solution (40) of these equations. In the coordinates (51)–(56), we
can use the $O(4)$ symmetry in order to diagonalize the equation for $\Gamma$.
First, we expand the fields in the spherical harmonics [47]

$\Phi(\tau,\omega)=\sum_{lm}Y_{lm}(\omega)\Phi_{lm}(\tau),$

where

$\triangle_{S}Y_{lm}=-l(l+2)Y_{lm}.$

$\triangle_{S}$ is the Laplace–Beltrami operator on $S^{3}$, $l$ is a natural
number, and $m=(j,\sigma)$ where $j=0,1,\ldots,l$ and $-j\leq\sigma\leq j$, is
the indexing solution of the Laplace–Beltrami operator with the eigenvalue
$-l(l+2)$ [48].

Then, we expand $\Gamma$ as

$\Gamma_{\tau}(\omega,\omega^{\prime})=\sum_{lm}Y_{lm}(\omega^{\prime})Y^{*}_{lm}(\omega)\Gamma_{lm}(\tau).$
(59)

Using Formula (24), we can define the Hamiltonian in each of the metrics
(51)–(58). Subsequently, solving Equation (25), we find a Gaussian solution
(40) of the Schrödinger equation in each of these coordinates. We obtain for
the metric (53) the Hamiltonian (24)

$\begin{array}[]{l}{\cal
H}=\sum_{lm}\Big{(}(H\cos(\tau))^{2}\Pi_{lm}^{2}+(H\cos(\tau))^{-2}l(l+2)\Phi_{lm}^{2}+M^{2}(H\cos(\tau))^{-4}\Phi_{lm}^{2}\Big{)}\end{array}$
(60)

defining for $t\geq 0$ the Schrödinger Equation (25).

The Hamiltonian for the Euclidean metric (56) in the interpretation (27) is
analogous to the upside-down oscillator [43, 42, 36] (we change
$M^{2}\rightarrow-\mu^{2}$; the Schrödinger equation is still
$i\hbar\partial_{\tau}\psi={\cal H}\psi$)

$\begin{array}[]{l}{\cal
H}=\sum_{lm}\Big{(}(H\cosh(\tau))^{2}\Pi_{lm}^{2}-(H\cosh(\tau))^{-2}l(l+2)\Phi_{lm}^{2}-\mu^{2}(H\cosh(\tau))^{-4}\Phi_{lm}^{2}\Big{)}\end{array}$
(61)

We still consider the Schrödinger Equation (26). In the Euclidean metric (56)
with the Hamiltonian (24) and the choice $\sqrt{-g}=-i\sqrt{|g|}$, we obtain
the diffusion Equation (26) for $\tau<0$ with

$\begin{array}[]{l}{\cal
H}_{E}=\sum_{lm}\Big{(}(H\cosh(\tau))^{2}\Pi_{lm}^{2}+(H\cosh(\tau))^{-2}l(l+2)\Phi_{lm}^{2}+\mu^{2}(H\cosh(\tau))^{-4}\Phi_{lm}^{2}\Big{)}\end{array}$
(62)

so that the Schrödinger equation takes the form

$\hbar\partial_{t}\psi={\cal H}_{E}\psi.$

The operators $\Gamma$ (28) and ${\cal G}$ (32) are diagonalized by an
expansion is spherical functions. We denote a function satisfying the wave
equation (34) for ${\cal G}$ by $u_{\tau}(\omega)$. Expanding
$u_{\tau}(\omega)$

$u(\tau,\omega)=\sum_{lm}Y_{lm}(\omega)u_{lm}(\tau)$

we obtain that in the metric (53) the coefficients $u_{lm}$ satisfy the
equation

$\partial_{\tau}^{2}u_{lm}+2\tan(\tau)\partial_{\tau}u_{lm}+(l(l+2)+M^{2}(H\cos(\tau))^{-2})u_{lm}=0.$
(63)

$\Gamma$ is related to $u$, as follows from Equation (32)

$u=\exp\Big{(}\int^{\tau}ds(H\cos(s))^{2}\Gamma(s)ds\Big{)}.$ (64)

For the Euclidean metric with $\sqrt{-g}\rightarrow\sqrt{|g|}$, the
corresponding formulas read (this equation also follows from the Hamiltonian
(61))

$\begin{array}[]{l}\partial_{\tau}^{2}u^{E}_{lm}-2\tanh(\tau)\partial_{\tau}u^{E}_{lm}-l(l+2)u^{E}_{lm}-\mu^{2}(H\cosh(\tau)^{-2}u^{E}_{lm}=0\end{array}$
(65)

Note that the definition $\sqrt{-g}\rightarrow-i\sqrt{|g|}$ does not change
the “wave equation” (65) for the inverted signature. With the inverted
signature, the “wave equation” becomes an elliptic equation, hence it does not
describe a wave propagation anymore.

$\Gamma^{E}$ is related to $u^{E}$ by

$u^{E}=\exp\Big{(}i\int^{\tau}ds(H\cosh(s))^{2}\Gamma^{E}(s)ds\Big{)}.$ (66)

Concerning the flat expanding metric of the general form

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\equiv dt^{2}-a^{2}(t)d{\bf x}^{2}$ (67)

introduced first for de Sitter in Equation (57) (and further discussed in
models of subsequent sections) we note that Friedmann equations governing the
evolution of $a(t)$

$(a^{-1}\frac{d}{dt}a)^{2}-\frac{1}{3}\Lambda=\frac{8\pi G}{3}\rho,$
$2a^{-1}\frac{d^{2}a}{dt^{2}}+(a^{-1}\frac{d}{dt}a)^{2}-\Lambda=-8\pi Gp$

are invariant under $a^{2}\rightarrow-a^{2}$. Here, $\Lambda$ is the
cosmological constant ($H=\sqrt{\frac{\Lambda}{3}}$), $\rho$ is the energy
density and $p$ is the pressure. This transformation can equivalently be
treated as $a\rightarrow ia$. The physical interpretation forces us to choose
as a solution the metric with $a^{2}>0$ (such a requirement may be not
applicable in quantum gravity).

For the homogeneous metric (67), owing to the translation invariance, we can
decompose $\Gamma$ in Fourier components

$\Gamma({\bf x}-{\bf y})=(2\pi)^{-3}\int d{\bf k}\Gamma({\bf k})\exp(i{\bf
k}({\bf x}-{\bf y})).$

We consider solutions satisfying the condition $\Gamma({\bf k})=\Gamma(-{\bf
k})=\Gamma(k)$, where $k=|{\bf k}|$. Then, in Fourier transform Equation (28)
in an expanding metric is

$\partial_{t}\Gamma+a^{-3}\Gamma^{2}+a{\bf k}^{2}+M^{2}a^{3}=0.$

We Fourier transform the wave function

$u({\bf x})=(2\pi)^{-\frac{3}{2}}\int d{\bf k}\exp(-i{\bf kx})u_{k}.$

Then, the wave Equation (34) is

$\partial_{t}^{2}u_{k}+3a^{-1}\partial_{t}a\partial_{t}u_{k}+a^{-2}k^{2}u_{k}+M^{2}u_{k}=0.$
(68)

The relation between $u$ and $\Gamma$ is determined by Equation (32)

$u_{k}=\exp(\int^{t}dsa^{-3}\Gamma_{s}(k))$ (69)

When $a^{-2}<0$ for $t<0$, we chose
$\sqrt{-g}=-i\sqrt{|g|}=-i|a^{2}|^{\frac{3}{2}}$. The Schrödinger equation for
$t>0$ is

$\begin{array}[]{l}i\hbar\partial_{t}\psi_{t}=\frac{1}{2}\int d{\bf
x}\Big{(}-\hbar^{2}a^{-3}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}+a(\nabla\Phi)^{2}+M^{2}a^{3}\Phi^{2}\Big{)}\psi_{t}\end{array}$ (70)

whereas for negative time, when $a^{2}<0$ and $\sqrt{-g}$ is imaginary, we
have the diffusion Equation (26) (as discussed in Section 10)

$\begin{array}[]{l}\hbar\partial_{t}\psi_{t}=\frac{1}{2}\int d{\bf
x}\Big{(}-\hbar^{2}|a^{2}|^{-\frac{3}{2}}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}+\sqrt{|a^{2}|}(\nabla\Phi)^{2}+\mu^{2}|a^{2}|^{\frac{3}{2}}\Phi^{2}\Big{)}\psi_{t}\end{array}$
(71)

where, for negative time, we changed the notation $M^{2}=-\mu^{2}$, suggesting
that $M^{2}$ should be chosen to be negative.

If in the Lagrangian (22) $\sqrt{-g}\rightarrow\sqrt{|g|}$, then the
Schrödinger equation reads

$\begin{array}[]{l}i\hbar\partial_{t}\psi_{t}=\frac{1}{2}\int d{\bf
x}\Big{(}-\hbar^{2}(|a^{2}|)^{-\frac{3}{2}}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}+a^{-2}(|a^{2}|)^{\frac{3}{2}}(\nabla\Phi)^{2}-\mu^{2}(|a^{2}|)^{\frac{3}{2}}\Phi^{2}\Big{)}\psi_{t}\end{array}$
(72)

Equation (72) is an analog of the one for an inverted oscillator (see [36], so
that the evolution is still unitary).

We note that from Equation (68) that it follows that the Wronskian is a
constant as

$\partial_{t}(a^{3}(u\partial_{t}u^{*}-u^{*}\partial_{t}u))=0$

For complex solutions, we choose the normalization

$u\partial_{t}u^{*}-u^{*}\partial_{t}u=-ia^{-3}.$ (73)

which is fixing the constant in canonical commutation relations. If
$a=\exp(Ht)$, as in Equation (57) then the (complex) solution of Equation (68)
is the cylinder function $Z_{\nu}$ [49, 50, 51, 52]

$u=a^{-\frac{3}{2}}Z_{\nu}(\frac{k}{H}\exp(-Ht)),$ (74)

where

$\nu=\frac{3}{2}\sqrt{1-\frac{4M^{2}}{9H^{2}}}.$

In general, as solutions of the wave equation we can take superpositions of
solutions with different $k$. The canonical quantization is realized with the
requirement [50] that, in the remote past $t_{0}\rightarrow-\infty$, the
solution tends to the plane wave (in conformal coordinates). Then,
$Z_{\nu}=H^{(2)}_{\nu}$, where $H^{(2)}_{\nu}$ is the Hankel function of the
second kind [49]. If $M=0$, then we can obtain real solutions of Equation (68)
$J_{\frac{3}{2}}(\frac{k}{H}\exp(-Ht))$ and
$Y_{\frac{3}{2}}(\frac{k}{H}\exp(-Ht))$, important for the construction of
interactions in Sections 11 and 12. These real solutions will give oscillatory
evolution kernels with caustic singularities.

The solution of Equation (68) with an inverted metric ($k\rightarrow ik$ and
$M\rightarrow i\mu$ in Equation (68)) is

$u=a^{-\frac{3}{2}}J_{\nu}(i\frac{k}{H}\exp(-Ht))=Ca^{-\frac{3}{2}}K_{\nu}(\frac{k}{H}\exp(-Ht))$

with a certain constant $C$, and

$\nu=\frac{3}{2}\sqrt{1+\frac{4\mu^{2}}{9H^{2}}}$

It can be seen that this is a solution of the Euclidean “wave equation”,
corresponding to the field theory on the Euclidean version of the anti-de
Sitter space [46, 53].

Summarizing, the aim of this section was to reduce the general Equations
(28)–(34) to a manageable form using the symmetry of the manifold. In the
homogeneous expanding coordinates, we can use the Fourier transform to
represent the operator $\Gamma$ as a multiplication operator in the Fourier
space. In angular coordinates (covering the whole of de Sitter space), we can
expand the solution in terms of spherical harmonics. A change in the spatial
signature in angular coordinates transforms de Sitter space into a sphere. The
Hamiltonian and the solution of the Schrödinger equation are expressed in
terms of a discrete set of variables. The quantization of these variables (as
outlined in Sections 2–4) is achieved by stochastic equations in the next
section.

##  6 Stochastic Equations for de Sitter Field and Fields in an Expanding Flat
Metric

In this and in subsequent sections, we discuss the stochastic time evolution
for positive, negative and imaginary time. Until now, only positive time was
considered in the stochastic representation (45), because the Brownian motion
is defined for a positive time. We can obtain a stochastic representation for
a negative time, taking the complex conjugation of Equation (25)

$i\hbar\partial_{-t}\psi_{t}^{*}={\cal
H}^{*}\psi_{t}^{*}\equiv\tilde{H}(-t)\psi^{*}_{t}.$ (75)

If the Schrödinger Equation (25) is to be defined for positive and negative
time, then the expressions for the Hamiltonian as a function of the metric
tensor must have a meaning in this range of time. This may be not possible if
Hawking–Penrose positivity conditions of the energy-momentum are to be
satisfied [32] (then the metric tensor may become degenerate and
$\frac{1}{\sqrt{-g}}$ infinite). The Einstein equations are invariant under
the time reflection. However, the reflected metric can violate the requirement
of $-g>0$, as will be discussed in Section 10. In such a case, $\tilde{{\cal
H}}(-t)$ for $t>0$ in the interpretation (24) does not define a Hermitian
operator. In fact, in Section 10, ${\cal H}(-t)$ will be anti-Hermitian. In
such a case, the Schrödinger equation for a positive time is transformed into
a diffusion equation for a negative time.

In general, on a globally hyperbolic manifold, if we find $\Gamma_{t}$ from
Equation (28), then the stochastic equation generated by the Hamiltonian (24)
reads (where ${\cal J}$ is defined in Equation (30))

$d\Phi_{s}=-{\cal
J}(t-s)\Gamma_{t-s}\Phi_{s}ds+\sqrt{i\hbar}\sqrt{g_{00}}(-g)^{-\frac{1}{4}}(t-s)dW_{s}.$
(76)

For a negative time, Equation (76) reads

$d\Phi_{s}={\cal
J}(t-s)\Gamma_{t-s}\Phi_{s}d(-s)+\sqrt{-i\hbar}\sqrt{g_{00}}(-g)^{-\frac{1}{4}}(t-s)dW_{-s}.$
(77)

With $\sqrt{-g}=-i\sqrt{|g|}$ and
$(-g)^{\frac{1}{4}}=\sqrt{-i}|g|^{\frac{1}{4}}$ in Equation (77) for $-g<0$.
Such a choice of square roots will give the real noise ($\sqrt{-i}$ cancels)
and a positive operator ${\cal H}^{E}$ in the diffusion Equation (26).

Equation (77) can be rewritten as

$d\Phi_{s}={\cal G}^{-1}\partial_{t}{\cal
G}(t-s)\Phi_{s}d(-s)+\sqrt{\hbar}\sqrt{g_{00}}|g|^{-\frac{1}{4}}(t-s)dW_{-s}.$
(78)

where ${\cal G}$ is the solution of the wave Equation (34) (with an inverted
metric). Equation (76) is a generalization of Equation (44) (considered in the
Minkowski space-time).

In the expanding metric (67), the Hamiltonian (24) for $t>0$ is

${\cal H}=\frac{1}{2}\int d{\bf
x}\Big{(}-\hbar^{2}a^{-3}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}+a(\nabla\Phi)^{2}+M^{2}a^{3}\Phi^{2}\Big{)}.$ (79)

The stochastic Equation (76) for $t\geq 0$ reads

$d\Phi_{s}=-a(t-s)^{-3}\Gamma(t-s)\Phi_{s}ds+a(t-s)^{-\frac{3}{2}}\sqrt{i\hbar}dW_{s}.$
(80)

By a differentiation of Equation (80), we obtain a random wave equation

$\begin{array}[]{l}(\partial_{s}^{2}-a^{-2}(t-s)\triangle+M^{2})\Phi_{s}+3a^{-1}(t-s)\partial_{t}a(t-s)\partial_{s}\Phi_{s}\cr=\Big{(}\frac{9}{2}a^{-1}(t-s)\partial_{t}a(t-s)-a^{-3}(t-s)\Gamma(t-s)\Big{)}\sqrt{i\hbar}\partial_{s}W\cr+\sqrt{i\hbar}a^{-\frac{3}{2}}(t-s)\partial^{2}_{s}W.\end{array}$
(81)

Equation (80) has the solution (with the initial condition $\Phi$ at $s=0$)

$\Phi_{s}=u_{t-s}u_{t}^{-1}\Phi+\sqrt{i\hbar}u_{t-s}\int_{0}^{s}u_{t-\tau}^{-1}a_{t-\tau}^{-\frac{3}{2}}dW_{\tau}.$
(82)

Equation (80) can be interpreted in the semi-classical approximation if in the
solution (3.20) $\Gamma$ is real (so $u$ is real). Then, in the limit
$\hbar\rightarrow 0$, Equation (80) reads

$\frac{d\Phi_{s}}{ds}=-a^{-3}(t-s)\Gamma(t-s)\Phi_{s}.$ (83)

Equation (83) relates the Hamilton–Jacobi limit of the Schrödinger equation
with its classical solution $\Phi_{s}$. In the standard formulation of the
Hamilton–Jacobi theory, if $S$ is the classical action, then the classical
trajectory is defined by [44]

$a^{-3}\frac{d\Phi_{s}}{ds}=\frac{\delta S}{\delta\Phi_{s}}.$

From Equation (82), the classical solution with the initial condition $\Phi$
is

$\Phi_{s}=u_{t-s}u_{t}^{-1}\Phi,$

where $u_{t-s}$ is the classical solution of the wave equation resulting from
the Lagrangian (22).

For the negative time, the stochastic evolution is a simple reflection if
$a(t)=a(-t)$, as appears in the effective field theories resulting from the
string theory [54, 55]. Then, we have $\tilde{{\cal H}}(-t)={\cal H}(t)$, and
$a$ is contracting to zero as $|t|\rightarrow 0$ and expanding to infinity
when $t\rightarrow\infty$. In such a case, the stochastic Equation (77) takes
the form

$d\Phi_{s}=\Gamma(s-t)a^{-3}(t-s)\Phi_{s}d(-s)+\sqrt{-i\hbar}a^{-\frac{3}{2}}(t-s)dW_{-s},$

where $a^{-3}\Gamma=u^{-1}\partial_{t}u$. It has the solution

$\Phi_{s}(\Phi)=u_{t-s}^{(-)}(u_{t}^{(-)})^{-1}-\sqrt{-i\hbar}u_{t-s}^{(-)}\int_{0}^{-s}(u_{t+\tau}^{(-)})^{-1}a_{t+\tau}^{-\frac{3}{2}}dW_{\tau},$
(84)

where $u_{t}^{(-)}$ is the solution of the wave Equation (68) for a negative
time. In this case, the field $\Phi_{s}$ can be considered as a simple
reflection of the one for $s>0$. We discuss an example of $a(t)\simeq|t|$ in
Section 10.

We return to de Sitter expanding metric (57). From Equation (69),

$\Gamma=a^{3}\frac{d}{dt}\ln\Big{(}a^{-\frac{3}{2}}Z_{\nu}\big{(}\frac{k}{H}\exp(-Ht)\big{)}\Big{)}.$

Hence,

$\begin{array}[]{l}\Phi_{s}=\exp(-\int_{t_{0}}^{s}(a^{-3}\Gamma)(t-\tau)d\tau)\Phi+\sqrt{i\hbar}\int_{t_{0}}^{s}\exp(-\int_{\tau}^{s}(a^{-3}\Gamma)(t-\tau^{\prime})d\tau^{\prime})a(t-\tau)^{-\frac{3}{2}}dW_{\tau}\end{array}$

or

$\begin{array}[]{l}\Phi_{s}=a(t-s)^{-\frac{3}{2}}Z_{\nu}(\frac{k}{H}\exp(-Ht+Hs))a(t-t_{0})^{\frac{3}{2}}\cr\Big{(}Z_{\nu}(\frac{k}{H}\exp(-Ht+Ht_{0}))\Big{)}^{-1}\Phi+\sqrt{i\hbar}a(t-s)^{-\frac{3}{2}}Z_{\nu}(\frac{k}{H}\exp(-Ht+Hs))\cr\int_{t_{0}}^{s}\Big{(}Z_{\nu}(\frac{k}{H}\exp(-Ht+H\tau))\Big{)}^{-1}dW_{\tau}.\end{array}$
(85)

The solution of the stochastic equation determines the field correlation
functions (from Equations (47) and (85), $\hat{\Phi}$ denotes the quantum
field, we set $t_{0}=0$)

$\begin{array}[]{l}(\psi_{0}^{g},\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g})=\int d\Phi|\psi_{t}^{g}(\Phi)|^{2}\Phi({\bf
k})E\Big{[}\Phi_{t}(\Phi,{\bf
k}^{\prime})\Big{]}\cr=i(\Gamma(t)-\Gamma(t)^{*})^{-1}Z_{\nu}(\frac{k}{H})a(t)^{\frac{3}{2}}\Big{(}Z_{\nu}(\frac{k}{H}\exp(-Ht))\Big{)}^{-1}\delta({\bf
k}+{\bf k}^{\prime}),\end{array}$ (86)

where we have used the covariance (from Equation (40))

$\int d\Phi|\psi_{t}^{g}(\Phi)|^{2}\Phi({\bf k})\Phi({\bf
k}^{\prime})=i\hbar(\Gamma-\Gamma^{*})^{-1}\delta({\bf k}+{\bf k}^{\prime}).$
(87)

In general,

$Z_{\nu}=\beta H_{\nu}^{(1)}+\alpha H_{\nu}^{(2)},$

where $H_{\nu}$ are the Hankel functions (in order to satisfy canonical
commutation relations we must have $|\alpha|^{2}-|\beta|^{2}=1$).

In order to calculate the rhs of Equation (87) we apply (with
$z=\frac{k}{H}\exp(-Ht)$)

$\begin{array}[]{l}\Gamma(t)-\Gamma(t)^{*}=a^{3}Hz(Z_{\nu}^{*}Z_{\nu})^{-1}\Big{(}\frac{d}{dz}Z_{\nu}^{*}Z_{\nu}-\frac{d}{dz}Z_{\nu}Z_{\nu}^{*}\Big{)}.\end{array}$
(88)

The rhs of the expression (87) may have any sign. For $\beta=0$ and $\alpha=1$
(the Bunch–Davies vacuum [52]), we have

$\Gamma-\Gamma^{*}=i\Big{(}H_{\nu}^{(2)*}H_{\nu}^{(2)}\Big{)}^{-1}$ (89)

Then, inserting Equation (89) in Equation (86), we obtain

$\begin{array}[]{l}(\psi_{0}^{g},\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g})=\hbar
H^{(2)}_{\nu}(\frac{k}{H})a(t)^{\frac{3}{2}}\Big{(}H^{(2)}_{\nu}(\frac{k}{H}\exp(-Ht)\Big{)}^{*}\delta({\bf
k}+{\bf k}^{\prime}).\end{array}$ (90)

From Equation (47), using Equation (90), we can also calculate (for the
massless field)

$\begin{array}[]{l}(\psi_{0}^{g},\hat{\Phi}_{t}({\bf k})\hat{\Phi}_{t}({\bf
k}^{\prime})\psi_{0}^{g})=(\psi_{t}^{g},\Phi({\bf k})\Phi({\bf
k}^{\prime})\psi_{t}^{g})\cr=\hbar\delta({\bf k}+{\bf
k}^{\prime})H_{\frac{3}{2}}^{(2)*}H_{\frac{3}{2}}^{(2)}(\frac{k}{H}\exp(-Ht))\end{array}$

and

$\begin{array}[]{l}(\psi_{0}^{g},\Phi({\bf k})\Phi({\bf
k}^{\prime})\psi_{0}^{g})-(\psi_{0}^{g},\hat{\Phi}_{t}({\bf
k})\hat{\Phi}_{t}({\bf k}^{\prime})\psi_{0}^{g})\cr=\hbar\delta({\bf k}+{\bf
k}^{\prime})\frac{1}{2k}(1-\exp(-2Ht))\end{array}$

a result obtained in [56, 57, 58, 59].

In a similar way, using Equation (47), we calculate higher order correlation
functions

$\begin{array}[]{l}(\psi_{0}^{g},\hat{\Phi}_{t}({\bf
k}_{1})\hat{\Phi}_{t}({\bf k}_{2})\hat{\Phi}({\bf
k}^{\prime}_{3})\hat{\Phi}({\bf k}_{4}^{\prime})\psi_{0}^{g})\cr=\int
d\Phi|\psi_{t}^{g}(\Phi)|^{2}\Phi({\bf k}_{1})\Phi({\bf
k}_{2})E\Big{[}\Phi_{t}(\Phi,{\bf k}^{\prime}_{3})\Phi_{t}(\Phi,{\bf
k}_{4}^{\prime})\Big{]}\end{array}$

In this equation, we insert the solution (82). The integral over $\Phi$ is the
Gaussian integral with the covariance (87). The expectation value in the
formula for the correlations is

$\begin{array}[]{l}E\Big{[}\Phi_{t}(\Phi,{\bf
k}^{\prime}_{3})\Phi_{t}(\Phi,{\bf k}_{4}^{\prime})\Big{]}\cr=(u_{0}\Phi)({\bf
k}^{\prime}_{3})((u_{t}\Phi)({\bf k}^{\prime}_{3}))^{-1}(u_{0}\Phi)({\bf
k}^{\prime}_{4})((u_{t}\Phi)({\bf k}^{\prime}_{4}))^{-1}\cr+i\hbar
u_{0}^{2}\delta({\bf k}^{\prime}_{3}+{\bf
k}^{\prime}_{4})\int_{0}^{t}u_{\tau}^{-2}({\bf
k}_{3}^{\prime})a_{\tau}^{-3}d\tau\equiv E_{cl}+G_{t},\end{array}$ (91)

where the $G_{t}$ term (it still will be discussed in subsequent sections)
being the quantum fluctuation of $\Phi_{t}$ is proportional to $\hbar$. The
integral (91) can explicitly be calculated for $M=0$. Then, inserting
$H_{\frac{3}{2}}^{(2)}$

$a_{\tau}^{\frac{3}{2}}u_{\tau}(k)=Cz^{-\frac{3}{2}}(z-i)\exp(-iz)$

with a certain constant $C$ and $z(\tau)=\frac{k}{H}\exp(-H\tau)$, we obtain
the integral in Equation (91). The integral reads

$\begin{array}[]{l}G_{t}=-i\hbar(\frac{k}{H}-i)^{2}\frac{H^{2}}{k^{3}}\exp(-2i\frac{k}{H})\cr\int_{\frac{k}{H}}^{\frac{k}{H}\exp(-Ht)}y^{2}(y-i)^{-2}\exp(2iy)dy.\end{array}$

For a small $k$, the leading infrared behavior of the real part of $G_{t}$ is
$\frac{Ht}{k}$ confirming the diffusive behavior of $\Phi_{t}$ discovered in
[56, 57, 58, 59].

In the case of an inverted metric (58) describing the Euclidean anti-de Sitter
space the correlation function is an analytic continuation of Equation (90)

$\begin{array}[]{l}H^{(2)}_{\nu}(i\frac{k}{H})a(t)^{-\frac{3}{2}}\Big{(}H^{(2)}_{\nu}(i\frac{k}{H}\exp(-Ht)\Big{)}^{*}\delta({\bf
k}+{\bf k}^{\prime})\cr\simeq
K_{\nu}(\frac{k}{H})a(t)^{-\frac{3}{2}}K_{\nu}(\frac{k}{H}\exp(-Ht)\Big{)}\delta({\bf
k}+{\bf k}^{\prime})\end{array}$ (92)

with $\nu=\frac{3}{2}\sqrt{1+\frac{4\mu^{2}}{9H^{2}}}$. The correlation
function (90) coincides with the one derived in [50, 52, 51] describing the
quantum free field in de Sitter space, with the ground state invariant under
the de Sitter group. The two-point function (92) describes the Euclidean
version of the quantum field on the anti-de Sitter space [6, 46].

With the solution (85)–(90), we shall have the same problem with a perurbative
construction of an interaction (ultraviolet divergencies and renormalization)
as in the QFT in the Minkowski space-time. In Sections 11 and 12, we discuss a
method to construct interactions when the solution of the wave equation $u(t)$
is a real function, and we do not insist on the existence of the ground state.
In the derivation of Equation (90), we have chosen as $u$ the Hankel function
$H_{\nu}^{(2)}(\frac{k}{H}\exp(-Ht))$. In such a case, $\Gamma$ is complex.
Hence, $\psi_{t}^{g}$ is not a pure phase WKB solution. If $M=0$, then
$\nu=\frac{3}{2}$. We could chose as $u$ the Bessel functions
$J_{\frac{3}{2}}$ or $Y_{\frac{3}{2}}$, which are expressed by trigonometric
functions. In such a case, $\Gamma$ is real and $\psi_{t}^{g}$ is a pure
phase. We obtain another representation of de Sitter field with caustic
singularities, which will be discussed in another model in Section 10. The
Bessel functions of an imaginary argument $K_{\nu}$ in Equation (92) give a
solution in Euclidean AdS. In such a case, $u_{\nu}$ is a real function
without caustic poles. We can construct interaction without an ultraviolet
cutoff by means of the Feynman–Kac formula, as discussed in Section 12.

Let us still determine the stochastic field (76) resulting from the
Hamiltonian (60). It is the solution of the stochastic equation

$d\Phi_{lm}(s)=-\partial_{\tau}\ln(u_{lm}(\tau-s))\Phi_{lm}(s)ds+\sqrt{i\hbar}H^{\frac{3}{2}}\cos(\tau-s)dw_{lm}(s),$
(93)

where $u_{lm}$ is the solution of Equation (63) and $w_{lm}$ are Gaussian
processes with mean zero and the covariance

$E[w_{lm}(s)w_{l^{\prime}m^{\prime}}(s^{\prime})]=\delta_{ll^{\prime}}\delta(m+m^{\prime})min(s,s^{\prime}),$
(94)

where $m=(j,\sigma)$ (as explained at Equation (59)) and
$\delta(m+m^{\prime})=\delta(j-j^{\prime})\delta(\sigma+\sigma^{\prime})$.

The Euclidean Hamiltonian (62) generates the stochastic equation

$\begin{array}[]{l}d\Phi_{lm}(s)^{E}=-\partial_{\tau}\ln(u^{E}_{lm}(\tau-s))\Phi^{E}_{lm}(s)d(-s)+\sqrt{\hbar}H^{\frac{3}{2}}\cosh(\tau-s)dw_{lm}(-s),\end{array}$
(95)

where $u^{E}_{lm}$ is the solution of Equation (65).

The solution defines a real diffusion process solving the diffusion Equation
(26) for $\tau\leq 0$.

The solution of Equation (93) is

$\begin{array}[]{l}\Phi_{lm}(s)=u_{lm}(\tau-s)u_{lm}(\tau)^{-1}\Phi_{lm}+\sqrt{i\hbar}H^{\frac{3}{2}}u_{lm}(\tau-s)\int_{0}^{s}(u_{lm}(\tau-t))^{-1}\cos(\tau-t)dw_{lm}(t).\end{array}$
(96)

Equation (95) has the solution

$\begin{array}[]{l}\Phi_{lm}^{E}(s)=u_{lm}^{E}(\tau-s)u^{E}_{lm}(\tau)^{-1}\Phi_{lm}+\sqrt{\hbar}u_{lm}^{E}(\tau-s)H^{\frac{3}{2}}\int_{0}^{s}(u_{lm}^{E}(\tau-t))^{-1}\cosh(\tau-t)dw_{lm}(t),\end{array}$
(97)

where $\Phi_{lm}$ is the initial condition at $s=0$. We need to calculate

$\begin{array}[]{l}E[(\Phi_{lm}(s)-E[\Phi_{lm}(s)])(\Phi_{lm}(s^{\prime})-E[\Phi_{l^{\prime}m^{\prime}}(s^{\prime})])]=G_{lm}(s,s^{\prime})\delta_{ll^{\prime}}\delta(m+m^{\prime}).\end{array}$

We have

$\begin{array}[]{l}G_{lm}(s,s^{\prime})=i\hbar
u_{lm}(\tau-s)u_{lm}(\tau-s^{\prime})H^{3}\int_{0}^{m(s,s^{\prime})}dt(u_{lm}(\tau-t))^{-2}\cos^{2}(\tau-t)\end{array}$
(98)

for the field (96) and

$\begin{array}[]{l}G^{E}_{lm}(s,s^{\prime})=\hbar
u^{E}_{lm}(\tau-s)u^{E}_{lm}(\tau-s^{\prime})H^{3}\int_{0}^{m(s,s^{\prime})}dt(u_{lm}^{E}(\tau+t))^{-2}\cosh^{2}(\tau-t)\end{array}$
(99)

for the inverted metric of Equation (56).

For a general $l$ and $M$, the solution of the wave equation is defined by the
Legendre functions [50]. We are unable to calculate the integrals in
$G_{lm}(s,s^{\prime})$ (98) and (99) exactly. For $l=0$ and $M=0$, we have

$u_{00}(\tau)=\tau+\frac{1}{4}\sin(2\tau)$ (100)

and

$u_{00}^{E}(\tau)=\tau+\frac{1}{4}\sinh(2\tau).$

For a large $l$, we can obtain the WKB solution (the odd solution) of the wave
equation

$u_{lm}(\tau)=(\frac{dS}{d\tau})^{-\frac{1}{2}}\sin(S(\tau))\cos(\tau),$ (101)

where

$S(s)=\int_{0}^{s}d\tau\sqrt{l(l+2)+1+(M^{2}-2H^{2})H^{-2}\cos^{-2}(\tau)}.$

In the even solution, we replace $\sin$ in Equation (101) by $\cos$. For the
Euclidean Equation (65), the WKB solution is of the form (101), but the
trigonometric functions are replaced by hyperbolic functions. So, the odd
solution reads

$u^{E}_{lm}(\tau)=(\frac{dS^{E}}{d\tau})^{-\frac{1}{2}}\sinh(S^{E}(\tau))\cosh(\tau)$
(102)

with

$S^{E}(s)=\int_{0}^{s}d\tau\sqrt{l(l+2)+1+(\mu^{2}-2H^{2})H^{-2}\cosh^{-2}(\tau)}.$
(103)

For the Euclidean version, an exponential solution of Equation (65) will be
needed (the sum of even and odd solutions)

$u_{lm}(\tau)=(\frac{dS^{E}}{d\tau})^{-\frac{1}{2}}\exp(S^{E}(\tau))\cosh(\tau).$
(104)

We can calculate $G_{lm}$ using the WKB approximation with the result

$\begin{array}[]{l}G_{lm}(s,s^{\prime})=i\hbar
H^{3}u_{lm}(\tau-s)u_{lm}(\tau-s^{\prime})(\cot(S(\tau-m(s,s^{\prime}))-\cot
S(\tau))\end{array}$ (105)

for $u_{lm}$ defined by Equation (101) and

$\begin{array}[]{l}G_{lm}(s,s^{\prime})=\hbar
H^{3}u^{E}_{lm}(\tau-s)u^{E}_{lm}(\tau-s^{\prime})(\coth(S^{E}(\tau-m(s,s^{\prime}))-\coth
S^{E}(\tau))\end{array}$ (106)

for the inverted metric in Equation (102).

In the case of the exponential solution (104), we obtain

$\begin{array}[]{l}G_{lm}(s,s^{\prime})=\hbar
H^{3}u^{E}_{lm}(\tau-s)u^{E}_{lm}(\tau-s^{\prime})(\exp(-2S^{E}(\tau-m(s,s^{\prime}))-\exp(-2S^{E}(\tau))).\end{array}$
(107)

For the even solutions of the WKB form defined by $\cos$ and $\cosh$ in
Equations (101) and (102), the cotangent functions in Equations (105) and
(106) are replaced by the tangents. The approximate expression for
$G_{lm}(s,s^{\prime})$ can be applied for calculations of the propagators and
field correlations in subsequent sections. In Appendix F, we express fields in
de Sitter space and their correlations in the cosmic time. There, we also
discuss de Sitter fields in two dimensions (see earlier papers [60, 61]),
where we express the correlation functions by elementary functions.

With the trigonometric functions entering $G_{lm}(s,s^{\prime})$, we have the
difficulty with the integrability over time in the definition of the evolution
kernel in Section 7 of quantum field theory in the Schrödinger formulation
(because of the poles of the trigonometric functions in Equation (105)). We
note that the replacement of the trigonometric functions by hyperbolic
functions (signature inversion) in the approximate WKB solutions (106) avoids
this difficulty. This analytic continuation allows a definition of the
Feynman–Kac formula for a real time (but with an inverted spatial signature),
as will be discussed in Sections 11 and 12.

##  7 The Propagator for the Schrödinger Evolution of the Scalar Field

We can express the time evolution either by the solution of the stochastic
equation as in Equation (45) or by an evolution kernel defined by

$(U_{t}\psi)(\Phi)=\int
d\Phi^{\prime}\tilde{K}_{t}(\Phi,\Phi^{\prime})\psi(\Phi^{\prime}).$

We write $\psi$ in the form (5). Then, the definition of the kernel is
rewritten as

$(U_{t}\psi_{0}^{g}\chi)(\Phi)=\psi_{t}^{g}\int
d\Phi^{\prime}K_{t}(\Phi,\Phi^{\prime})\chi(\Phi^{\prime}).$ (108)

We represent $\chi$ as a Fourier transform

$\chi(\Phi)=\int d\Lambda\tilde{\chi}(\Lambda)\exp(i(\Lambda,\Phi)).$

Then, from Equation (45), we obtain

$\begin{array}[]{l}K_{t}(\Phi,\Phi^{\prime})=\int d\Lambda
E\Big{[}\exp\Big{(}i(\Lambda,\Phi_{t}(\Phi)-\Phi^{\prime})\Big{)}\Big{]}.\end{array}$
(109)

We calculate the expectation value of $\Phi_{t}$ with the result

$\begin{array}[]{l}E\Big{[}\exp\Big{(}i(\Lambda,\Phi_{t}(\Phi)-\Phi^{\prime})\Big{)}\Big{]}=\exp\Big{(}i(\Lambda,u_{0}u_{t}^{-1}\Phi-\Phi^{\prime})-\frac{1}{2}(\Lambda,G_{t}\Lambda)\Big{)}.\end{array}$
(110)

Then, calculating the $\Lambda$ integral we obtain (up to a normalization
constant)

$\begin{array}[]{l}K_{t}(\Phi,\Phi^{\prime})=\det
G_{t}^{-\frac{1}{2}}\exp\Big{(}-\frac{1}{2}\Big{(}(u_{0}u_{t}^{-1}\Phi-\Phi^{\prime}),G_{t}^{-1}(u_{0}u_{t}^{-1}\Phi-\Phi^{\prime})\Big{)}\Big{]},\end{array}$
(111)

where

$\begin{array}[]{l}G_{t}({\bf k},{\bf k}^{\prime})=E[(\Phi_{t}({\bf
k})-<\Phi_{t}>({\bf k}))(\Phi_{t}({\bf k}^{\prime})-<\Phi_{t}>({\bf
k}^{\prime}))]\cr\equiv G_{t}(k)\delta({\bf k}+{\bf k}^{\prime})\end{array}$

with $<\Phi>=E[\Phi]$.

In a homogeneous expanding space-time for positive time (to be concise, from
now on we omit the $\delta({\bf k}+{\bf k}^{\prime})$ term in $G_{t}({\bf
k},{\bf k}^{\prime})$)

$G_{t}=i\hbar u_{0}^{2}\int_{0}^{t}u_{t-\tau}^{-2}a_{t-\tau}^{-3}d\tau.$ (112)

Equation (112), (where $m(s,s^{\prime})\equiv min(s,s^{\prime})$), is a
consequence of

$E\Big{[}\int_{0}^{s}f_{\tau}dW_{\tau}({\bf
k})\int_{0}^{s^{\prime}}f_{\tau^{\prime}}dW_{\tau^{\prime}}({\bf
k}^{\prime})\Big{]}=\int_{0}^{m(s,s^{\prime})}f^{2}_{\tau}d\tau\delta({\bf
k}+{\bf k}^{\prime})$

following from the definition of the stochastic integral [38].

In the presence of an interaction from Equation (46), we obtain a
generalization of the Formula (111) for the evolution kernel as

$\begin{array}[]{l}K_{t}(\Phi,\Phi^{\prime})=\int d\Lambda
E\Big{[}\exp\Big{(}i(\Lambda,\Phi_{t}(\Phi)-\Phi^{\prime})\Big{)}\exp\Big{(}-\frac{i}{\hbar}\int_{0}^{t}V(\Phi_{s}(\Phi))ds\Big{)}\Big{]}.\end{array}$

Clearly, we cannot explicitly calculate this expectation value and the
$\Lambda$ integral as we did in Equations (110) and (111), but we can perform
such calculations in a perturbation expansion in $V$, as will be discussed in
Sections 11 and 12.

For negative time in Equation (78), the correlation function is

$G_{t}=i\hbar
u_{0}^{(-)}u_{0}^{(-)}\int_{0}^{-t}(u_{t+\tau}^{(-)})^{-2}a_{t+\tau}^{-3}d\tau$
(113)

where $u^{(-)}$ is the solution of the wave equation for a negative time. Let
us note that, for positive time, we obtain an oscillating propagator $K_{t}$,
which is a pure phase, whereas for an inverted metric (for the negative time),
we obtain a real function describing a transition function for a diffusion.

The result of (111) and (112) gives an explicit formula for the evolution
kernel. Another way to calculate this propagator involves a solution of the
Cauchy problem for the wave equation, as discussed briefly at the end of
Appendix D (such calculations are performed in more detail in [62]). Note that
a real $u$ leads to a purely imaginary $G_{t}$. Then, we obtain the propagator
(111), which is a pure phase in agreement with the Feynman Formula (36) and
Appendix D. If $u$ is complex, then $\Gamma$ is complex. Hence, $G_{t}$ is
complex. Such a modification of the kernel results from its definition (108)
involving $\psi_{t}^{g}$, which is not a pure phase.

Using the expectation value (110), we can also calculate the field correlation
functions in a state $\psi_{0}^{g}\chi$ with a Gaussian state $\chi$ of the
form

$\chi=\det B^{\frac{1}{2}}\exp(-\frac{1}{2}(\Phi,B\Phi)).$ (114)

Then, from Equation (110) (up to a constant multiplier)

$\begin{array}[]{l}\chi_{t}(\Phi)=\det B\int
d\Lambda\exp(-\frac{1}{2}(\Lambda,B^{-1}\Lambda))\cr\exp\Big{(}i(\Lambda,u_{0}u_{t}^{-1}\Phi)-\frac{1}{2}(\Lambda,G_{t}\Lambda)\Big{)}=\det(1+BG_{t})^{-\frac{1}{2}}\cr\det
B^{\frac{1}{2}}\exp\Big{(}-\frac{1}{2}(u_{0}u_{t}^{-1}\Phi,(B^{-1}+G_{t})^{-1}u_{0}u_{t}^{-1}\Phi)\Big{)}.\end{array}$
(115)

Next,

$\begin{array}[]{l}(\Phi\chi)_{t}(\Phi)=u_{0}u_{t}^{-1}B^{-1}(B^{-1}+G_{t})^{-1}\Phi\det
B^{\frac{1}{2}}\det(1+BG_{t})^{-\frac{1}{2}}\cr\exp\Big{(}-\frac{1}{2}(u_{0}u_{t}^{-1}\Phi,(B^{-1}+G_{t})^{-1}u_{0}u_{t}^{-1}\Phi)\Big{)}.\end{array}$
(116)

Then, the field correlation function according to Equation (48) can be
calculated (for a general solution $\psi_{t}^{g}$ of the Schrödinger Equation
(25)) from the formula

$\begin{array}[]{l}(\psi_{0}^{g}\chi,\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g}\chi)=\delta({\bf k}+{\bf k}^{\prime})\int
d\Phi|\psi_{t}^{g}|^{2}\chi^{*}_{t}(\Phi)\Phi({\bf k})(\Phi\chi)_{t}(\Phi,{\bf
k}^{\prime}),\end{array}$ (117)

where $\chi_{t}$ and $(\Phi\chi)_{t}$ have been evaluated in Equations (115)
and (116).

The integral (117) is Gaussian. Hence, we derive an explicit (although quite
complicated) formula for the correlation function. The formula simplifies if,
e.g., $|\psi_{t}^{g}|^{2}$ is integrable (when $i(\Gamma-\Gamma^{*})<0$) and
$\chi=1$ (as in Equation (91)). Then,
$(\Phi\chi)_{t}(\Phi)=E[\Phi_{t}(\Phi)]=u_{0}u_{t}^{-1}\Phi$.

We can calculate the field correlations in various states
$(\psi_{0}^{g}\chi,\hat{\Phi}(t,{\bf x})\hat{\Phi}({\bf
x}^{\prime})\psi_{0}^{g}\chi)$ using the propagator (111) as

$\begin{array}[]{l}\int d\Phi
d\Phi^{\prime}d\Phi^{\prime\prime}\Big{(}U_{t}(\Phi,\Phi^{\prime})(\psi_{0}^{g}\chi)(\Phi^{\prime})\Big{)}^{*}\Phi({\bf
x})U_{t}(\Phi,\Phi^{\prime\prime})\Phi^{\prime\prime}({\bf
x}^{\prime})(\psi_{0}^{g}\chi)(\Phi^{\prime\prime})\cr=\int d\Phi
d\Phi^{\prime}d\Phi^{\prime\prime}|\psi_{t}^{g}(\Phi)|^{2}K_{t}(\Phi,\Phi^{\prime})^{*}K_{t}(\Phi,\Phi^{\prime\prime})\chi(\Phi^{\prime})^{*}\chi(\Phi^{\prime\prime})\Phi({\bf
x})\Phi^{\prime\prime}({\bf x}^{\prime})\end{array}$ (118)

with

$U_{t}(\Phi,\Phi^{\prime})=\psi_{t}^{g}(\Phi)K_{t}(\Phi,\Phi^{\prime})(\psi_{0}^{g}(\Phi^{\prime}))^{-1}.$
(119)

We can express all multi-time correlation functions by the evolution kernel.
As an example,

$\begin{array}[]{l}(\psi_{0}^{g}\chi,\hat{\Phi}_{t}({\bf
x})\hat{\Phi}_{t^{\prime}}({\bf
x}^{\prime})\psi_{0}^{g}\chi)=(U_{t}\psi_{0}^{g}\chi,\Phi({\bf
x})U_{t,t^{\prime}}\Phi({\bf x}^{\prime})\psi_{0}^{g}\chi)\end{array}$ (120)

where

$U_{t,t^{\prime}}(\Phi,\Phi^{\prime})=\psi_{t}^{g}(\Phi)K_{(t,t^{\prime})}(\Phi,\Phi^{\prime})(\psi_{t^{\prime}}^{g}(\Phi^{\prime}))^{-1}.$
(121)

Hence, in terms of the kernels Equation (120) is expressed as

$\begin{array}[]{l}\int d\Phi
d\Phi^{\prime}d\Phi^{\prime\prime}\Big{(}U_{t}(\Phi,\Phi^{\prime})(\psi_{0}^{g}\chi)(\Phi^{\prime})\Big{)}^{*}\Phi({\bf
x})U_{t,t^{\prime}}(\Phi,\Phi^{\prime\prime})\Phi^{\prime\prime}({\bf
x}^{\prime})(\psi_{0}^{g}\chi)(\Phi^{\prime\prime})=\cr\int d\Phi
d\Phi^{\prime}d\Phi^{\prime\prime}|\psi_{t}^{g}(\Phi)|^{2}K_{t}(\Phi,\Phi^{\prime})^{*}K_{(t,t^{\prime})}(\Phi,\Phi^{\prime\prime})\chi(\Phi^{\prime})^{*}\chi(\Phi^{\prime\prime})\Phi({\bf
x})\Phi^{\prime\prime}({\bf x}^{\prime})\end{array}$ (122)

The kernel $K_{(t,t^{\prime})}$ is obtained as in Equation (109)

$K_{(t,t^{\prime})}(\Phi,\Phi^{\prime})=(\det
G(t,t^{\prime}))^{-\frac{1}{2}}\exp\Big{(}-\frac{1}{2}(u_{0}(u_{t-{t}^{\prime}})^{-1}\Phi-\Phi^{\prime})G(t,t^{\prime})^{-1}(u_{0}(u_{t-t^{\prime}})^{-1}\Phi-\Phi^{\prime})\Big{)},$
(123)

where now the process $\Phi_{s}$ is the solution of the stochastic equation
with an initial value at $t^{\prime}$ (instead of zero)

$\Phi_{s}=u_{t-s}(u_{t-t^{\prime}})^{-1}\Phi+\sqrt{i\hbar}u_{t-s}\int_{t^{\prime}}^{s}u_{t-\tau}^{-1}a_{t-\tau}^{-\frac{3}{2}}dW_{\tau}$
(124)

Hence, from Equations (109) and (110),

$G(t,t^{\prime})=i\hbar
u_{0}^{2}\int_{t^{\prime}}^{t}u_{t-\tau}^{-2}a_{t-\tau}^{-3}d\tau$ (125)

Similarly to Equation (122) in terms of the kernel
$U_{t,t^{\prime}}(\Phi,\Phi^{\prime})$, we can express the $n$-point functions

$(\psi_{0}^{g}\chi,\hat{\Phi}_{t_{1}}({\bf
x}_{1})\ldots\ldots\hat{\Phi}_{t_{n}}({\bf
x}_{n})\psi_{0}^{g}\chi)=(U_{t_{1}}\psi_{0}^{g}\chi,\Phi({\bf
x}_{1})U_{t_{1},t_{2}}\Phi({\bf x}_{2})\ldots\ldots U_{t_{n}-t_{n-1}}\Phi({\bf
x}_{n})\psi_{0}^{g}\chi)$ (126)

In the presence of interaction, there are additional Feynman–Kac factors in
$K_{(t,t^{\prime})}$ as in Equation (48)

$\exp\Big{(}-\frac{i}{\hbar}\int^{t}_{t^{\prime}}V(\Phi_{s})ds\Big{)}$

At the end of this section, we would like to discuss different notions of
propagators in the literature [63, 64] (the one for the metric $a\simeq t$ has
been discussed in [65, 66]). In these papers, the propagator has been defined
as the inverse of the operator ${\cal A}$ appearing in the action (22) when
written in the form $\int dxL=\int dx\Phi{\cal A}\Phi$. We could also
represent this propagator by Schwinger’s proper time

${\cal A}^{-1}=i\int_{0}^{\infty}d\tau\exp(-i\tau{\cal A}).$

By formal functional integration, the functional integral average
$<\phi(x)\phi(x^{\prime})>$ is equal to ${\cal A}^{-1}(x,x^{\prime})$(the
kernel satisfying ${\cal A}{\cal A}^{-1}=1$). In general, by formal
differentiation of the lhs of Equation (117) using Equation (23) in the metric
(67), we obtain for $t>0$ under the assumption that the quantum field
satisfies the wave equation

$\begin{array}[]{l}\partial_{t}^{2}(\psi_{0}^{g}\chi,\hat{\Phi}_{t}({\bf
k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g}\chi)=\partial_{t}(\psi_{0}^{g}\chi,a^{-3}\Pi_{t}({\bf
k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g}\chi)\cr=(-3H\partial_{t}-a^{-2}k^{2}-M^{2})(\psi_{0}^{g}\chi,\Phi_{t}({\bf
k})\Phi({\bf k}^{\prime})\psi_{0}^{g}\chi)\delta({\bf k}+{\bf
k}^{\prime}).\end{array}$

Hence, the correlation function also satisfies the wave equation. If $\chi=1$
and $|\psi_{t}^{g}|^{2}$ is integrable, then

$\begin{array}[]{l}(\psi_{0}^{g},\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g})=\int d\Phi|\psi_{t}^{g}|^{2}\Phi({\bf
k})E[\Phi_{t}({\bf k}^{\prime})]\cr=u_{0}({\bf k})u_{t}^{*}({\bf
k})\delta({\bf k}+{\bf k}^{\prime})\end{array}$

because $E[\Phi_{t}({\bf k}^{\prime})]=(u_{0}u_{t}^{-1}\Phi)({\bf k})$ (from
Equation (82)) and $i(\Gamma-\Gamma^{*})\simeq(u_{t}u_{t}^{*})^{-1}$ from the
Wronskian (as has been exploited in the particular case of the de Sitter
metric (57) in Equation (90)). We can conclude that, although the correlation
function in any state is a solution of the wave equation, then the solution of
the equation for the correlation functions is not unique, because it depends
on the state under consideration. If there is a unique ground state (as in de
Sitter space) invariant under a symmetry group, then we can distinguish the
solution having this invariance [51]. In other states, the two-point
correlation function must be determined through calculations, e.g., from
Equation (117) by means of the propagator (111) (as will be discussed at the
end of Section 9).

In QFT in the Minkowski space-time in the ground state (11), we obtain ${\cal
A}^{-1}$ as the Lorentz invariant Green function of
$\partial_{t}^{2}-\triangle+M^{2}$ equal to
$\frac{1}{2}\nu^{-1}\exp(-i\nu|t-t^{\prime}|)$, where
$\nu=\sqrt{-\triangle+M^{2}}$. In the case of the de Sitter space-time, the
calculation with $\chi=1$ and $\psi^{g}$ defined by $\Gamma$ in Equation (90)
coincides (as discussed after Equation (117)) with the result of an
expectation value computed either by a formal functional integration or
derived by an expansion of the field in creation and annihilation operators
defined by de Sitter invariant vacuum [50, 51, 47]. In the time-dependent
metric of this section (as well as in Section 8), there is no candidate for a
vacuum. Hence, it remains unclear in which state $\psi_{0}^{g}\chi$ the two-
point correlation function could be equal to ${\cal A}^{-1}$. If there is no
unique vacuum for the quantum field in an expanding metric, then the physical
meaning of ${\cal A}^{-1}$ is obscure, whereas the propagator defined in this
section (and the correlation functions defined by the propagator) has a clear
meaning for the canonically quantized field theory of Section 3. A relation of
the propagator $K_{t}$ to ${\cal A}^{-1}$, defined as the causal propagator,
has been discussed in ref. [62]. In this paper, the definition of the
propagator $K_{t}$ is related to the solution of the Cauchy problem, as
expressed by the causal propagator. We briefly discuss the method of the
calculation of $K_{t}$ using the solution of the Cauchy problem for ${\cal A}$
at the end of Appendix D.

##  8 Power-Law Expansion

In this and subsequent sections, we discuss some soluble wave equations. We
are unable to derive an explicit solution of the wave Equation (68) in a
homogeneous space-time for an arbitrary expansion $a(t)$. In Appendix B, we
solve this equation for a large $k$ by means of the WKB method (it can be made
exact as in [63]). Then, in this approximation, we can calculate the
propagator (111) and the field correlation functions (Appendix B).

In this and subsequent sections, we discuss soluble models. First, we consider
the model with the power-law expansion $a^{2}=b_{0}^{-2}(t+\gamma)^{2\alpha}$
with a general $\alpha\in R$ and $t+\gamma>0$ (we shift the initial time by
$\gamma$ in order to pose the initial condition at $t=0$, even if $\gamma=0$
corresponds to a degenerate metric). If $t+\gamma<0$, then $a$ is a complex
function in general (but even powers of $t$ lead to admissible models with a
topology change between positive and negative time [67]). We discuss the case
$\alpha=\frac{1}{2}$ for $t+\gamma<0$ in Section 10. The expansion law
$a=b_{0}^{-1}|t+\gamma|^{\alpha}$ for both positive $t+\gamma$ and negative
$t+\gamma$ appears in cosmological models resulting from string theory [54,
55]. Such a metric cannot be a solution of general relativity, because it is
not differentiable at $t+\gamma=0$.

The wave equation for $t+\gamma>0$ has the form

$\partial_{t}^{2}u+3\alpha(t+\gamma)^{-1}\partial_{t}u+b_{0}^{2}(t+\gamma)^{-2\alpha}k^{2}u+M^{2}u=0.$
(127)

The corresponding stochastic equation is

$d\Phi_{s}=-u^{-1}(t-s)\partial_{t}u(t-s)ds+\sqrt{i\hbar}a^{-\frac{3}{2}}dW_{s}.$

For $t+\gamma<0$ and $a=b_{0}^{-1}|t+\gamma|^{\alpha}$ (in string inspired
models [54, 55]), the wave equation is

$\partial_{t}^{2}u+3\alpha(t+\gamma)^{-1}\partial_{t}u+b_{0}^{2}|t+\gamma|^{-2\alpha}k^{2}u+M^{2}u=0.$

Equation (127) is explicitly soluble if $M=0$ ([68], 2.162 Equation (1a)). The
solution is (for $t+\gamma>0$, we may choose here either complex-valued or
real-valued Bessel functions)

$u=(t+\gamma)^{\frac{1-3\alpha}{2}}Z_{\nu}(\frac{b_{0}}{1-\alpha}k(t+\gamma)^{1-\alpha}),$
(128)

where

$\nu=\frac{1-3\alpha}{2(1-\alpha)}.$

$\alpha$ is related to $w$ in the equation of state ($p=w\rho$)

$\alpha=\frac{2}{3(1+w)}.$

Note that $\nu=-\frac{1}{2}$ if $\alpha=\frac{1}{2}$ (radiation,
$w=\frac{1}{3}$) and $\nu=-\frac{3}{2}$ if $\alpha=\frac{2}{3}$ (dust, $w=0$).
We discuss $\alpha=\frac{1}{2}$ in detail in Section 10.

Let us briefly consider the interesting case of $\alpha=\frac{2}{3}$. Then, a
real solution valid for both $t+\gamma>0$, as well as $t+\gamma<0$, which is
continuous at $t+\gamma=0$, is expressed by elementary functions
($T=3b_{0}(t+\gamma)^{\frac{1}{3}}$ is the conformal time)

$\begin{array}[]{l}u=(t+\gamma)^{-\frac{2}{3}}\Big{(}\cos(3b_{0}k(t+\gamma)^{\frac{1}{3}})-(3b_{0}k(t+\gamma)^{\frac{1}{3}})^{-1}\sin(3b_{0}k(t+\gamma)^{\frac{1}{3}})\Big{)}.\end{array}$

The real solutions $u$ are distinguished in our approach as they lead to real
$\Gamma$ and purely imaginary $G_{t}$ in Equation (112). This property is
relevant for a construction of interactions via the Feynman–Kac formula (see
Section 11). We also consider a complex solution, which gives a square
integrable $\psi_{t}^{g}$

$u=(t+\gamma)^{-\frac{1}{2}}H^{(2)}_{\frac{3}{2}}(3b_{0}k(t+\gamma)^{\frac{1}{3}})$

The two-point function in this state is discussed at the end of this section.

For a general $\nu$ from Equation (69), we obtain

$\Gamma=(t+\gamma)^{3\alpha}\frac{d}{dt}\ln\Big{(}(t+\gamma)^{\frac{1}{2}-\frac{3\alpha}{2}}Z_{\nu}(\frac{b_{0}}{1-\alpha}k(t+\gamma)^{1-\alpha})\Big{)}.$

The stochastic Equation (80) reads

$\begin{array}[]{l}d\Phi_{s}=-\frac{d}{dt}\ln\Big{(}(t+\gamma-s)^{\frac{1}{2}-\frac{3\alpha}{2}}Z_{\nu}(\frac{b_{0}}{1-\alpha}k(t+\gamma-s)^{1-\alpha})\Big{)}\Phi_{s}ds+\sqrt{i\hbar}(t+\gamma-s)^{-\frac{3\alpha}{2}}dW_{s}.\end{array}$
(129)

The solution is

$\begin{array}[]{l}\Phi_{s}=\Big{(}(t+\gamma-s)^{\frac{1}{2}-\frac{3\alpha}{2}}Z_{\nu}(\frac{b_{0}}{1-\alpha}k(t+\gamma-s)^{1-\alpha})\Big{)}\cr\Big{(}(t+\gamma-
t_{0})^{\frac{1}{2}-\frac{3\alpha}{2}}Z_{\nu}(\frac{1}{1-\alpha}k(t+\gamma-
t_{0})^{1-\alpha})\Big{)}^{-1}\Phi\cr+\sqrt{i\hbar}\Big{(}(t+\gamma-s)^{\frac{1}{2}-\frac{3\alpha}{2}}Z_{\nu}(\frac{b_{0}}{1-\alpha}k(t+\gamma-s)^{1-\alpha})\Big{)}\cr\times\int_{t_{0}}^{s}d\tau\Big{(}(t+\gamma-\tau)^{\frac{1}{2}-\frac{3\alpha}{2}}Z_{\nu}(\frac{b_{0}}{1-\alpha}k(t+\gamma-\tau)^{1-\alpha})\Big{)}^{-1}(t+\gamma-\tau)^{-\frac{3}{2}\alpha}dW_{\tau}.\end{array}$
(130)

Let

$z(t+\gamma)=\frac{b_{0}k}{1-\alpha}(t+\gamma)^{1-\alpha}.$

The general complex solution which can give a normalizable $\psi_{t}^{g}$ is a
superposition of Hankel functions. The Wronskian for the Hankel functions is
(here $H^{(2)}=H^{(1)*}$)

$(\frac{d}{dz}H^{(2)})H^{(1)}-(\frac{d}{dz}H^{(1)})H^{(2)}=-\frac{4i}{\pi z}$

Hence, for $Z_{\nu}=H^{(1)}_{\nu}$, we obtain

$\Gamma-\Gamma^{*}=\frac{4i}{\pi}(H^{(2)}_{\nu}H^{(2)*}_{\nu})^{-1}(t+\gamma)^{3\alpha}H$
(131)

Now, $\Gamma-\Gamma^{*}$ of Equation (131) leads to a normalizable Gaussian
state (allowing a computation of correlation functions, an analog of Equation
(90))

$\begin{array}[]{l}(\psi_{0}^{g},\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g})=\int d\Phi|\psi_{t}^{g}(\Phi)|^{2}\Phi
E\Big{[}\Phi_{t}(\Phi)\Big{]}\cr=\frac{\pi\hbar}{4H}H^{(2)}_{\nu}(z(\gamma))H^{(2)*}_{\nu}(z(t+\gamma))(t+\gamma)^{1-\frac{3}{2}\alpha}\delta({\bf
k}+{\bf k}^{\prime}).\end{array}$

We can express the correlations in the case of dust $\alpha=\frac{2}{3}$,
$\nu=-\frac{3}{2}$ by elementary functions. The two-point function at small
$k$ behaves as

$k^{-3}\exp(iz(t+\gamma)-iz(\gamma))$

It has the infrared singularity. $G_{t}$ is complex but the small $k$
expansion contains a real part

$\Re G_{t}\simeq\frac{t}{k}\delta({\bf k}+{\bf k}^{\prime})$

similar to the one in de Sitter space.

The random fields defined by the Hankel function $H_{\nu}^{(2)}$ do not allow
definition of the interaction by means of the Feynman–Kac integral in Sections
11 and 12. We can establish the Feynman integral with the choice $J_{\nu}$ or
$Y_{\nu}$ as $Z_{\nu}$. In such a case, after solving the Schrödinger equation
with an interaction in Section 11, we must look for states which are square
integrable and define the correlation functions by means of Equation (48)
(these could be the Gaussian states of Equation (116)).

##  9 The Expansion $a^{2}(t)=\frac{\epsilon}{b_{0}^{2}}(t+\gamma)^{2}$

This is the limiting case ($\alpha\rightarrow 1$) of Equation (127). It
describes the $w=-\frac{1}{3}$ fluid corresponding to the coasting cosmology
[69] or Dirac–Milne cosmology [70]. The scale $a(t)$ is invariant under
$t+\gamma\rightarrow-t-\gamma$) (it is contracting for $t+\gamma<0$ and
expanding for $t+\gamma>0$; we choose $\gamma>0$ for $t>0$ and $\gamma<0$ for
$t<0$). The scalar field theory of these models has been also discussed in
[65, 66]. As noticed in Section 5, the Einstein equations which can appear in
the path integral of quantum gravity together with $a^{2}\simeq t^{2}$ give
also the solution with $a^{2}\simeq-t^{2}$ (with the same $w=-\frac{1}{3}$).
The model with $a^{2}<0$ is interesting for the construction of an interaction
via the Feynman–Kac formula, as will be discussed in Section 12.

The wave equation reads (with $M=0$)

$\frac{d^{2}u}{dt^{2}}+3(t+\gamma)^{-1}\frac{du}{dt}+\epsilon(t+\gamma)^{-2}b_{0}^{2}k^{2}u=0.$
(132)

In Equation (132), we add a parameter $\epsilon=\pm 1$ in order to describe a
model with an inverted spatial metric resulting from the solution of Friedmann
equations $a^{2}=-\frac{1}{b_{0}^{2}}(t+\gamma)^{2}$ with $w=-\frac{1}{3}$.
Choosing $\sqrt{-g}=i\sqrt{|g|}$, we obtain a diffusion equation for positive
time instead of the Schrödinger equation. The diffusion equation makes sense
only in one direction of time (either positive or negative). Our choice of the
square root of $\sqrt{-g}$ in this section gives the diffusion equation for a
positive time. The diffusion Equation (26) reads

$\begin{array}[]{l}\hbar\partial_{t}\psi_{t}=\frac{1}{2}\int d{\bf
x}\Big{(}\hbar^{2}b_{0}^{3}|t+\gamma|^{-3}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}-\frac{|t+\gamma|}{b_{0}}(\nabla\Phi)^{2}-\mu^{2}b_{0}^{-3}|t+\gamma|^{3}\Phi^{2}\Big{)}\psi_{t}\end{array}$
(133)

The equation for $\chi$ is

$\partial_{t}\chi_{t}=\int d{\bf
x}\Big{(}\hbar\frac{1}{2}b_{0}^{-3}|t+\gamma|^{-3}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}-u^{-1}\partial_{t}u\Phi({\bf x})\frac{\delta}{\delta\Phi({\bf
x})}\Big{)}\chi_{t},$ (134)

where $u$ is the solution of Equation (132) with $\epsilon=-1$. We could
consider the model of Section 3, where there is $d{\bf x}\sqrt{|g|}$ as the
volume element. In such a case, we still have the Schrödinger Equation
(instead of the diffusion equation). Then, the equation for $\chi$ is

$\partial_{t}\chi_{t}=\int d{\bf
x}\Big{(}\hbar\frac{i}{2}b_{0}^{3}|t+\gamma|^{-3}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}-u^{-1}\partial_{t}u\Phi({\bf x})\frac{\delta}{\delta\Phi({\bf
x})}\Big{)}\chi_{t},$

where $u$ satisfies Equation (132) with $\epsilon=-1$ . We could also treat
the introduction of $\epsilon$ in Equation (132) as a technical step for a
derivation of solutions, which subsequently are to be continued analytically
to $\epsilon=1$. There is no continuous transition between $\epsilon=-1$ and
$\epsilon=1$.

The solution of Equation (132) is [68] (true for $t+\gamma>0$ as well as for
$t+\gamma<0$)

$u=C_{1}|t+\gamma|^{-1-\mu}+C_{2}|t+\gamma|^{-1+\mu},$ (135)

where

$\mu=\sqrt{1-\epsilon b_{0}^{2}k^{2}}.$ (136)

The conformal time is

$T=\int dta^{-1}=b_{0}\ln(t+\gamma).$

Hence, the classical solution in conformal time

$u=C_{1}\exp(-b_{0}^{-1}T(\mu+1))+C_{2}\exp(-b_{0}^{-1}T(-\mu+1))$

for large momenta (with $\epsilon=1$) $\mu\simeq ikb_{0}$ looks like a free
wave (with a decaying amplitude).

We choose $C_{1}=0$. Then,

$i\Gamma=-|t|^{3}u^{-1}\partial_{t}u=-t^{2}(\mu-1)<0.$

Hence, $\psi_{t}^{g}$ (40) is square integrable.

The solution of the stochastic Equation (80) for the model (134) is

$\Phi_{s}=u_{t-s}u_{t}^{-1}\Phi+\sqrt{\hbar}u_{t-s}b_{0}^{\frac{3}{2}}\int_{0}^{s}u_{t-\tau}^{-1}(t+\gamma-\tau)^{-\frac{3}{2}}dW_{\tau}.$
(137)

We are interested in an explicit calculation of the evolution kernel and
correlation functions. In such calculations, we need to evaluate

$\begin{array}[]{l}G_{s,s^{\prime}}({\bf k},{\bf
k}^{\prime})=E[(\Phi_{s^{\prime}}({\bf k})-<\Phi_{s^{\prime}}>({\bf
k}))(\Phi_{s}({\bf k}^{\prime})-<\Phi_{s}>({\bf k}^{\prime}))]\equiv
G_{s,s^{\prime}}(k)\delta({\bf k}+{\bf k}^{\prime}),\end{array}$

where $<\Phi>\equiv E[\Phi]$.

The operator $\mu=\sqrt{1+\epsilon b_{0}^{2}\triangle}$ with $\epsilon=1$
cannot be defined in the infinite dimensional setting. For this reason, we
consider the model with the inverted signature ($\epsilon=-1$)
$\triangle\rightarrow-\triangle$ when $\mu$ is a self-adjoint positive
operator in $L^{2}(d{\bf x})$. When $C_{1}=0$ then, with the inverted
signature, the operator
$u_{t-s}u_{t}^{-1}=\frac{t+\gamma}{t+\gamma-s}(\frac{t+\gamma-s}{t+\gamma})^{\mu}$
is well-defined as a contractive semigroup acting upon $\Phi$ in Equation
(137).

We have (with $C_{1}=0$ and $t+\gamma>0$) the following for Model (134):

$\begin{array}[]{l}G_{s,s^{\prime}}(k)=\hbar
u_{t-s}u_{t-s^{\prime}}\int_{0}^{m(s,s^{\prime})}d\tau
u_{t-\tau}^{-2}(t+\gamma-\tau)^{-3}\cr=\frac{\hbar}{2\mu}(t-s+\gamma)^{-1+\mu}(t-s^{\prime}+\gamma)^{-1+\mu}\cr\Big{(}(t+\gamma-m(s,s^{\prime}))^{-2\mu}-(t+\gamma)^{-2\mu}\Big{)},\end{array}$
(138)

where $m(s,s^{\prime})=min(s,s^{\prime})$.

It can be checked that the rhs of Equation (138) is expressed by a well-
defined operator $\exp(-r\mu)$ ($\epsilon=-1$), where $r\geq 0$.

For equal time in Model (138), we have

$\begin{array}[]{l}G_{t}=\frac{\hbar}{2\mu}\gamma^{-2}\Big{(}1-(1+\frac{t}{\gamma})^{-2\mu}\Big{)}\end{array}$
(139)

The limit $\gamma=0$ is infinite expressing the degeneracy of the metric at
$t+\gamma=0$. The evolution kernel is defined by $G_{t}$ in Equation (111). We
can express the correlation function of Equation (138) by the two-point
function $G^{E}_{M}$ of the scalar free field with a mass M using the formula
($\nu=\sqrt{k^{2}+M^{2}}$)

$\begin{array}[]{l}(2\pi)^{-3}\int d{\bf k}\exp(i{\bf
kx})(2\nu)^{-1}\exp(-s\nu)\cr=(2\pi)^{-4}\int dk_{0}d{\bf k}\exp(i{\bf
kx}+ik_{0}s)(k_{0}^{2}+{\bf k}^{2}+M^{2})^{-1}=G^{E}_{M}(s,{\bf
x})\end{array}$ (140)

It can be seen from Equations (138) and (140) that the random field
$\Phi_{s}({\bf k})$ has correlation function with the same large $k$ behavior
(the same short distance behavior) as the quantum Euclidean free field (after
the signature inversion). This holds true for all stochastic fields defined in
this paper. In the construction of polynomial interactions (Sections 11 and
12), and in the expansion of non-polynomial interactions in powers of the
fields [71], we shall have the same ultraviolet singularity and the same
renormalization problem as in the conventional Fock space approach or in the
constructive Euclidean framework. However, in the standard approach to QFT in
four dimensions, we cannot go beyond the perturbative framework because after
a renormalization the Feynman–Kac factor becomes unbounded. In non-polynomial
interactions the superpropagator [71] becomes extremely singluar. This can
change in the stochastic approach (as discussed in Sections 11 and 12),
because we can work with (bounded) Feynman–Kac oscillatory factors (in
particular, the superpropagator becomes an oscillatory hence Lebesgue
integrable function).

##  10 The Radiation Background $a^{2}(t)=c_{0}^{-1}(t+\gamma)$

In this section, we consider the massless scalar field with
$\alpha=\frac{1}{2}$ in Equation (132). We insert $\gamma\geq 0$ for $t\geq 0$
and $\gamma\leq 0$ for $t\leq 0$, so that when $\gamma\neq 0$, the metric is
not degenerate. The metric (for $t+\gamma>0$) is the solution of the Friedmann
equation for radiation with the energy density $\rho=\rho_{0}a^{-4}$ and the
pressure $p=\frac{1}{3}\rho$. It is usually rejected at $t+\gamma<0$ because
the inverted signature has no classical meaning ([72], Section 112), as it
violates the local special relativity principles. The inverted metric can
appear as a stationary point in quantum gravity, defined as an average over
the metric tensor. The causal structure in quantum gravity can disagree with
the classical one at the Planck scale.

We are interested in the behavior of the quantum scalar field evolution for
the metric $a^{2}(t)=c_{0}^{-1}(t+\gamma)$ for positive as well as negative
time in the limit $\gamma=0$. The Gaussian solution (40) is determined by a
solution of the equation

$\frac{d^{2}u}{dt^{2}}+\frac{3}{2}(t+\gamma)^{-1}\frac{du}{dt}+(t+\gamma)^{-1}c_{0}k^{2}u=0$
(141)

true for $t+\gamma>0$, as well as for $t+\gamma<0$.

For $t+\gamma<0$ we choose $\sqrt{-g}=-i\sqrt{|g|}$. With such a choice of the
square root, the diffusion Equation (26) reads

$\begin{array}[]{l}\hbar\partial_{t}\psi_{t}=\frac{1}{2}\int d{\bf
x}\Big{(}-\hbar^{2}c_{0}^{\frac{3}{2}}|t+\gamma|^{-\frac{3}{2}}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}\cr+c_{0}^{-\frac{1}{2}}|t+\gamma|^{\frac{1}{2}}(\nabla\Phi)^{2}-M^{2}c_{0}^{-\frac{3}{2}}|t+\gamma|^{\frac{3}{2}}\Phi^{2}\Big{)}\psi_{t}\end{array}$
(142)

The equation for $\chi$ is

$\partial_{t}\chi_{t}=\int d{\bf
x}\Big{(}-\hbar\frac{1}{2}c_{0}^{\frac{3}{2}}|t+\gamma|^{-\frac{3}{2}}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}-u^{-1}\partial_{t}u\Phi({\bf x})\frac{\delta}{\delta\Phi({\bf
x})}\Big{)}\chi_{t}.$ (143)

Equations (142) and (143) are well-defined for $t+\gamma<0$, because the
generator of the diffusion has a positively definite second order differential
operator.

We could consider the metric $g_{lk}=\delta_{lk}c_{0}^{-1}|t+\gamma|$, which
does not have the second derivative at $t+\gamma=0$. For this reason, it
cannot be a solution of Einstein equations, but appears as a solution in the
effective field theory resulting from the string theory [54]. For this metric,
the Schrödinger Equation (25) reads

$\partial_{t}\chi_{t}=\frac{1}{2}\int d{\bf
x}\Big{(}i\hbar\frac{1}{2}c_{0}^{\frac{3}{2}}|t+\gamma|^{-\frac{3}{2}}\frac{\delta^{2}}{\delta\Phi({\bf
x})^{2}}-u^{-1}\partial_{t}u\Phi({\bf x})\frac{\delta}{\delta\Phi({\bf
x})}\Big{)}\chi_{t},$

where $u$ is the solution of the equation

$\frac{d^{2}u}{dt^{2}}+\frac{3}{2}(t+\gamma)^{-1}\frac{du}{dt}+|t+\gamma|^{-1}c_{0}k^{2}u=0.$
(144)

In order to obtain a solution of Equation (144), it is useful to change the
cosmic time $t$ into the conformal time $T$ as
$T=2c_{0}^{\frac{1}{2}}\sqrt{t}$ for $t>0$ and
$T=-2c_{0}^{\frac{1}{2}}\sqrt{-t}$ for $t<0$ (then $a^{2}(T)\simeq T^{2}$ as
in [73]). Inserting $\tilde{u}=Tu$ into Equation (144), we can see that
$\tilde{u}$ satisfies the oscillator equation. Hence, the solution of Equation
(144) for $t+\gamma>0$ is a superposition of plane waves

$u=A_{1}T^{-1}\exp(ikT)+A_{2}T^{-1}\exp(-ikT).$

It follows that the solution of Equation (144) for positive as well as
negative time is

$\begin{array}[]{l}u_{t}=C_{1}|t+\gamma|^{-\frac{1}{2}}\cos(2k\sqrt{c_{0}}\sqrt{|t+\gamma}|)+C_{2}|t+\gamma|^{-\frac{1}{2}}\sin(2k\sqrt{c_{0}}\sqrt{|t+\gamma|})\end{array}$
(145)

We obtain a different solution of Equation (141). We express it by real
functions (for $t+\gamma>0$)

$\begin{array}[]{l}u_{t}=C_{1}(t+\gamma)^{-\frac{1}{2}}\cos(2k\sqrt{c_{0}}\sqrt{t+\gamma})+C_{2}(t+\gamma)^{-\frac{1}{2}}\sin(2k\sqrt{c_{0}}\sqrt{t+\gamma})\end{array}$
(146)

If $t+\gamma<0$, then the solution of Equation (141) is

$\begin{array}[]{l}u_{t}^{(-)}=C_{1}(-t-\gamma)^{-\frac{1}{2}}\cosh(2k\sqrt{c_{0}}\sqrt{-t-\gamma})+C_{2}(-t-\gamma)^{-\frac{1}{2}}\sinh(2k\sqrt{c_{0}}\sqrt{-t-\gamma})\end{array}$
(147)

For $t+\gamma<0$ Equation (141) (with $k^{2}\rightarrow-\triangle$) is an
elliptic equation. It does not describe waves.

The limit $\gamma\rightarrow 0$ of $u_{t}$ exists for all $|t|\geq 0$ (see a
discussion of continuity in [67, 74, 75, 76]) only if $C_{1}=0$. Then, the
limit $t+\gamma\rightarrow 0$ for positive time, as well as for the negative
time, is equal to $u_{0}=2C_{2}\sqrt{c_{0}}k$. The limit $t+\gamma\rightarrow
0$ of $\partial_{t}u_{t}$ also exists from both sides

$(\partial_{t}u_{t})_{|t=0}=-\frac{4}{3}C_{2}\sqrt{c_{0}}c_{0}k^{3}.$

The solution of the stochastic Equation (80) for $t\geq s\geq 0$ is

$\begin{array}[]{l}\Phi_{s}(\Phi)=u_{t-s}u_{t}^{-1}\Phi+\sqrt{i\hbar}u_{t-s}c_{0}^{\frac{3}{4}}\int_{0}^{s}u_{t-\tau}^{-1}|t+\gamma-\tau|^{-\frac{3}{4}}dW_{\tau}.\end{array}$
(148)

For $-t\geq-s\geq 0$ of the (string) metric, $a(t)\simeq|t|^{\frac{1}{2}}$ the
field $\Phi_{s}$ determined by Equations (77) and (84) is

$\begin{array}[]{l}\Phi_{s}(\Phi)=u_{t-s}u_{t}^{-1}\Phi+\sqrt{-i\hbar}u_{t-s}c_{0}^{\frac{3}{4}}\int_{0}^{-s}u_{t-\tau}^{-1}|t+\gamma+\tau|^{-\frac{3}{4}}dW_{\tau}.\end{array}$
(149)

The quantum field theory depends (for a positive time) on the correlation
function

$\begin{array}[]{l}G_{ss^{\prime}}=E[(\Phi_{s}({\bf k})-E[\Phi_{s}]({\bf
k}))(\Phi_{s^{\prime}}({\bf k}^{\prime})-E[\Phi_{s^{\prime}}]({\bf
k}^{\prime}))]=\delta({\bf k}+{\bf k}^{\prime})i\hbar
c_{0}^{\frac{3}{2}}u_{t-s}u_{t-s^{\prime}}\cr\times\int_{0}^{m(s,s^{\prime})}d\tau
u_{t-\tau}^{-2}|t+\gamma-\tau|^{-\frac{3}{2}}\equiv\delta({\bf k}+{\bf
k}^{\prime})G_{ss^{\prime}}(k).\end{array}$ (150)

where we denote $m(s,s^{\prime})\equiv min(s,s^{\prime})$ if $t\geq s\geq 0$
and $t\geq s^{\prime}\geq 0$. The $\delta({\bf k}+{\bf k}^{\prime})$ term in
Equation (150) will be omitted in the formulas below.

For a negative time of $a^{2}\simeq t$ in the model (141)–(143), there is no
$\sqrt{i}$ in Equation (82), which is canceled by the $\sqrt{i}$ factor in
$a^{-\frac{3}{2}}$. Hence, the counterpart of Equation (149) for the negative
time leads to the diffusion process $\Phi_{s}$ solving the diffusion Equation
(143)

$\begin{array}[]{l}\Phi_{s}(\Phi)=u^{(-)}_{t-s}(u^{(-)}_{t})^{-1}\Phi+\sqrt{\hbar}u^{(-)}_{t-s}c_{0}^{\frac{3}{4}}\int_{0}^{-s}(u_{t-\tau}^{(-)})^{-1}|t+\gamma+\tau|^{-\frac{3}{4}}dW_{\tau},\end{array}$

where, by $u^{(-)}_{t}$, we denote the solution (147) of the “wave equation”
(141) for the negative time. The correlation functions of the fields for a
negative time are determined by the formula

$\begin{array}[]{l}G_{ss^{\prime}}=E[(\Phi_{s}({\bf k})-E[\Phi_{s}]({\bf
k}))(\Phi_{s^{\prime}}({\bf k}^{\prime})-E[\Phi_{s^{\prime}}]({\bf
k}^{\prime}))]\cr=\delta({\bf k}+{\bf k}^{\prime})\hbar
c_{0}^{\frac{3}{2}}u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}\cr\times\int_{0}^{m(s,s^{\prime})}d\tau(u^{(-)}_{t-\tau})^{-2}|t+\gamma+\tau|^{-\frac{3}{2}}\equiv\delta({\bf
k}+{\bf k}^{\prime})G_{ss^{\prime}}(k),\end{array}$ (151)

where, for the negative time $m(s,s^{\prime})\equiv min(-s,-s^{\prime})$, if
$-t\geq-s\geq 0$ and $-t\geq-s^{\prime}\geq 0$.

As can be seen from Equation (143), for a negative time, $\Phi_{s}$ becomes a
real diffusion process and $K_{t}$ in Equation (111) is a real transition
function. The Schrödinger evolution equation (together with the Feynman–Kac
formula) takes the form of an evolution of the diffusion process. The reason
for this is the purely imaginary value of $\sqrt{-g}$. If
$a(t)\simeq|t+\gamma|^{\frac{1}{2}}$ then the expression for $\Phi_{s}$ and
$G_{ss^{\prime}}$ for negative time resembles the ones for the positive time
(for a negative time it is a reflection of the one for a positive time).

We can express the integrals (150) and (151) by elementary functions if
$C_{2}=0$, $C_{1}=0$, $C_{1}=\pm C_{2}$ and $C_{1}=\pm iC_{2}$.

If $C_{2}=0$, then we have, for $t+\gamma>0$,

$u_{t}=(t+\gamma)^{-\frac{1}{2}}\cos(2\sqrt{c_{0}}k\sqrt{t+\gamma}).$ (152)

If $C_{1}=0$, then

$u_{t}=(t+\gamma)^{-\frac{1}{2}}\sin(2\sqrt{c_{0}}k\sqrt{t+\gamma}).$ (153)

If $-t-\gamma>0$, then

$u^{(-)}_{t}=(-t-\gamma)^{-\frac{1}{2}}\cosh(2\sqrt{c_{0}}k\sqrt{-t-\gamma})$
(154)

and

$u^{(-)}_{t}=(-t-\gamma)^{-\frac{1}{2}}\sinh(2\sqrt{c_{0}}k\sqrt{-t-\gamma}).$
(155)

When $C_{1}=\pm C_{2}$, then

$u_{t}^{(-)}=(-t-\gamma)^{-\frac{1}{2}}\exp(\pm
2\sqrt{c_{0}}k\sqrt{-t-\gamma}).$ (156)

For the solution (152), we obtain

$\begin{array}[]{l}G_{ss^{\prime}}=i\hbar
u_{t-s}u_{t-s^{\prime}}c_{0}^{\frac{3}{2}}\int_{0}^{m(s,s^{\prime})}d\tau|t+\gamma-\tau|^{-\frac{1}{2}}(\cos(2\sqrt{c_{0}}k\sqrt{t+\gamma-\tau}))^{-2}\cr=i\hbar
u_{t-s}u_{t-s^{\prime}}c_{0}\frac{1}{k}(\tan(2\sqrt{c_{0}}k\sqrt{t-m(s,s^{\prime})+\gamma})-\tan(2\sqrt{c_{0}}k\sqrt{t+\gamma}))\end{array}$
(157)

where

$u_{0}=\frac{1}{\sqrt{\gamma}}\cos(2\sqrt{c_{0}}k\sqrt{\gamma}).$ (158)

The limit $\gamma\rightarrow 0$ of $u_{t}$ does not exist at $t=0$. Then, the
evolution kernel (111) is not defined. For the solution (153),

$\begin{array}[]{l}G_{ss^{\prime}}=i\hbar
u_{t-s}u_{t-s^{\prime}}c_{0}^{\frac{3}{2}}\int_{0}^{m(s,s^{\prime})}d\tau|t+\gamma-\tau|^{-\frac{1}{2}}(\sin(2\sqrt{c_{0}}k\sqrt{t+\gamma-\tau}))^{-2}\cr=i\hbar
u_{t-s}u_{t-s^{\prime}}c_{0}\frac{1}{k}(\cot(2\sqrt{c_{0}}k\sqrt{t+\gamma})-\cot(2\sqrt{c_{0}}k\sqrt{t-m(s,s^{\prime})+\gamma}))\end{array}$
(159)

The limit $\gamma\rightarrow 0$ of $u_{0}$ in Equation (153) is
$2\sqrt{c_{0}}k$. When $t\rightarrow 0$ and $\gamma\rightarrow 0$ then
$m(s,s^{\prime})\rightarrow 0$ and $G_{ss^{\prime}}\rightarrow 0$ in Equation
(159).

For the solution (154), we obtain at $t+\gamma<0$ in Equation (151)

$\begin{array}[]{l}G_{ss^{\prime}}=\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}^{\frac{3}{2}}\int_{0}^{m(s,s^{\prime})}d\tau|t+\gamma+\tau|^{-\frac{1}{2}}(\cosh(2\sqrt{c_{0}}k\sqrt{-t-\gamma-\tau}))^{-2}\cr=\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}\frac{1}{k}(\tanh(2\sqrt{c_{0}}k\sqrt{-t-m(s,s^{\prime})-\gamma})\cr-\tanh(2\sqrt{c_{0}}k\sqrt{-t-\gamma})).\end{array}$
(160)

$u_{0}$ has no limit when $\gamma\rightarrow 0$. Hence, the propagator (111)
cannot be defined in this limit.

For $u^{(-)}_{t}$ of Equation (155), we have

$\begin{array}[]{l}G_{ss^{\prime}}=\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}^{\frac{3}{2}}\int_{0}^{m(s,s^{\prime})}d\tau|t+\gamma+\tau|^{-\frac{1}{2}}(\sinh(2\sqrt{c_{0}}k\sqrt{-t-\gamma-\tau}))^{-2}\cr=\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}\frac{1}{k}(\coth(2\sqrt{c_{0}}k\sqrt{-t-\gamma})\cr-\coth(2\sqrt{c_{0}}k\sqrt{-t-m(s,s^{\prime})-\gamma}))\end{array}$
(161)

The limit $\gamma\rightarrow 0$ of $u_{0}$ in Equation (161) together with
Equation (111) also defines the evolution kernel in the limit
$\gamma\rightarrow 0$. There is an apparent singularity as $k\rightarrow 0$ in
the correlation $G_{ss^{\prime}}(k)$, but this singularity is canceled by the
volume element $d{\bf k}$ in the definition of the evolution kernel (111).

$G_{ss^{\prime}}$ (161) is decaying for a large $k$ as

$\begin{array}[]{l}G_{ss^{\prime}}\simeq\hbar\frac{1}{2}c_{0}(-t+s-\gamma)^{-\frac{1}{2}}(-t+s^{\prime}-\gamma)^{-\frac{1}{2}}k^{-1}\cr\Big{(}\exp(-4\sqrt{c_{0}}k\sqrt{-t-m(s,s^{\prime})-\gamma}+2\sqrt{c_{0}}k\sqrt{-t-\gamma+s}\cr+2\sqrt{c_{0}}k\sqrt{-t-\gamma+s^{\prime}})\cr-\exp(-4\sqrt{c_{0}}k\sqrt{-t-\gamma}+2\sqrt{c_{0}}k\sqrt{-t-\gamma+s}\cr+2\sqrt{c_{0}}k\sqrt{-t-\gamma+s^{\prime}})\Big{)}\end{array}$

The correlation functions of the field $\Phi_{s}({\bf x})$ can be expressed
(according to Equation (150) as the Fourier transform of $G_{ss^{\prime}}$.
From the large $k$ behavior of (161) and Equation (140), we can conclude that
the short distance behavior of the correlations of $\Phi_{s}({\bf x})$ is the
same as in the Euclidean free field theory of the scalar field.

The exponential solutions define the correlation functions

$\begin{array}[]{l}G_{ss^{\prime}}=\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}^{\frac{3}{2}}\int_{0}^{m(s,s^{\prime})}d\tau|t+\gamma+\tau|^{-\frac{1}{2}}(\exp(-2\sqrt{c_{0}}k\sqrt{-t-\gamma-\tau}))^{-2}\cr=\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}\frac{1}{2k}(\exp(4\sqrt{c_{0}}k\sqrt{-t-m(s,s^{\prime})-\gamma})\cr-\exp(4\sqrt{c_{0}}k\sqrt{-t-\gamma}))\end{array}$
(162)

for the minus sign in (156) and

$\begin{array}[]{l}G_{ss^{\prime}}=i\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}^{\frac{3}{2}}\cr\times\int_{0}^{m(s,s^{\prime})}d\tau|t+\gamma+\tau|^{-\frac{1}{2}}(\exp(2k\sqrt{-t-\gamma-\tau}))^{-2}\cr=i\hbar
u^{(-)}_{t-s}u^{(-)}_{t-s^{\prime}}c_{0}\frac{1}{2k}(\exp(-4\sqrt{c_{0}}k\sqrt{-t-\gamma})\cr-\exp(-4\sqrt{c_{0}}k\sqrt{-t-m(s,s^{\prime})-\gamma}))\end{array}$
(163)

for the plus sign. There is no limit $\gamma\rightarrow 0$ of $u_{0}$.

At the end of this section, let us consider some superpositions of the
solutions (152) and (153) with complex coefficients. So for positive time, let
us consider (for a negative time the solution of the “wave equation” (141) is
given by Equation (147))

$u_{t}=(t+\gamma)^{-\frac{1}{2}}\exp(2i\sqrt{c_{0}}k\sqrt{t+\gamma}).$ (164)

Calculating the field correlations, we obtain

$\begin{array}[]{l}G_{ss^{\prime}}=\frac{\hbar}{2}\frac{1}{\sqrt{c_{0}}k}\frac{1}{\sqrt{t+\gamma-s}}\frac{1}{\sqrt{t+\gamma-s^{\prime}}}\exp(2i\sqrt{c_{0}}k(\sqrt{t+\gamma-s}+\sqrt{t+\gamma-s^{\prime}}))\cr\Big{(}\exp(-4i\sqrt{c_{0}}k\sqrt{t+\gamma-m(s,s^{\prime}})-\exp(-4i\sqrt{c_{0}}k\sqrt{t+\gamma})\Big{)}\end{array}$
(165)

For $G_{t}$ in the propagator (111), we have

$\begin{array}[]{l}G_{t}=-\frac{1}{2\gamma\sqrt{c_{0}}k}\Big{(}\exp(-4i\sqrt{c_{0}}k\sqrt{t+\gamma})+4i\sqrt{c_{0}}k\sqrt{\gamma})-1\Big{)}\cr=-\frac{1}{2\gamma\sqrt{c_{0}}k}\Big{(}\cos(-4\sqrt{c_{0}}k\sqrt{t+\gamma})+4\sqrt{c_{0}}k\sqrt{\gamma})-1\cr-i\sin(-4\sqrt{c_{0}}k\sqrt{t+\gamma})+4\sqrt{c_{0}}k\sqrt{\gamma})\Big{)}\end{array}$
(166)

Note that $\Re G_{t}>0$. Hence, the function defining
$K_{t}(\Phi,\Phi^{\prime})$ in Equation (111) is integrable (this would not be
so if, instead of $u_{t}$ in Equation (164), we have considered $u_{t}^{*}$).

With a complex $u_{t}$, the solution $\psi_{t}^{g}$ (40) of the Schrödinger
equation is not a phase factor, and it may grow to infinity for a large $\Phi$
(this would be so for $u_{t}^{*}$). Let us calculate from Equation (69) for
$u_{t}$ of Equation (164)

$i\Gamma=a^{3}u^{-1}\partial_{t}u=-\sqrt{c_{0}}k(t+\gamma)-\frac{i}{2}\sqrt{t+\gamma}.$
(167)

Hence, $\Re(i\Gamma)<0$, showing that $\psi_{t}^{g}$ is square integrable for
the complex solution (164) for positive time. The correlation function is

$\begin{array}[]{l}(\psi_{0}^{g}\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g})=\gamma^{-\frac{1}{2}}\exp(2i\sqrt{c_{0}}k\sqrt{\gamma})(t+\gamma)^{\frac{1}{2}}\exp(-2i\sqrt{c_{0}}k\sqrt{t+\gamma})\cr\times(2\sqrt{c_{0}}k(t+\gamma))^{-1}\delta({\bf
k}+{\bf k}^{\prime})\end{array}$

Then, for a small time,

$\begin{array}[]{l}(\psi_{0}^{g},\Phi({\bf k})\Phi({\bf
k}^{\prime})\psi_{0}^{g})-(\psi_{0}^{g}\hat{\Phi}_{t}({\bf k})\hat{\Phi}({\bf
k}^{\prime})\psi_{0}^{g})\simeq\Big{(}t(4\sqrt{c_{0}}k\gamma^{2})^{-1}+it\gamma^{-\frac{3}{2}}\Big{)}\delta({\bf
k}+{\bf k}^{\prime})\end{array}$

This real and imaginary parts of the diffusive (linear in $t$) behavior of
$\Phi^{2}$ can also be seen from Equation (165).

For the negative time with the solution

$u^{(-)}_{t}=(-t-\gamma)^{-\frac{1}{2}}\exp(2\sqrt{c_{0}}k\sqrt{-t-\gamma})$
(168)

$\psi_{t}^{g}$ is also square integrable, as
$i\Gamma=(a^{2})^{\frac{3}{2}}u^{(-)}\partial_{t}u^{(-)}<0$. For the solutions
(164) and (168) in the free field theory in the radiation background, we can
calculate correlation functions using the Formula (47), as we did in the case
of the de Sitter background in Equations (90) and (91). However, with these
solutions, $u_{t}$ it is difficult to define the interaction via the
Feynman–Kac formula, because we are unable to prove that the Feynman–Kac
factor is a bounded function (as discussed in [36] and in Sections 11 and 12).

We may consider more general superpositions of solutions of wave Equation
(141) by an addition of a piece with negative frequency to Equation (164) for
a positive time

$u_{t}=(t+\gamma)^{-\frac{1}{2}}(\exp(2i\sqrt{c_{0}}k\sqrt{t+\gamma})+(\alpha+i\beta)\exp(-2i\sqrt{c_{0}}k\sqrt{t+\gamma})$
(169)

and

$u^{(-)}_{t}=(-t-\gamma)^{-\frac{1}{2}}(\exp(2\sqrt{c_{0}}k\sqrt{-t-\gamma})+(\alpha+i\beta)\exp(-2\sqrt{c_{0}}k\sqrt{-t-\gamma}).$
(170)

for a negative time. We can calculate $\Gamma=a^{3}u^{-1}\partial_{t}u$ for a
positive time and
$\Gamma=-i(|a^{2}|)^{\frac{3}{2}}(u^{(-)})^{-1}\partial_{t}u^{(-)}$ for a
negative time. We did not find square integrable solutions ($\Re(i\Gamma)<0$),
except for the cases (164) and (168), leading to a square integrable wave
function for a quantum scalar field in a radiation background. The limit
$\gamma\rightarrow 0$ of the degenerate metric does not exist from both sides
of time except of the solutions (153) and (155).

Let us summarize the results of this section. When we choose real solutions
$u$, then, for a positive time (when $-g>0$), $\Gamma$ is real. Hence,
$\psi_{t}^{g}$ is a pure phase. $G_{t}$ (112) (as well as $G_{ss^{\prime}}$)
is purely imaginary. For negative time, $a$ is purely imaginary
$a=ic_{0}^{-\frac{1}{2}}\sqrt{|t+\gamma|}$. Then,

$i\Gamma=c_{0}^{-\frac{3}{2}}|t+\gamma|^{\frac{3}{2}}u_{t}^{(-)}\partial_{t}u_{t}^{(-)}$
(171)

is a real function. We have checked, using Equation (155), that $i\Gamma$ is
negative (hence $\psi_{t}^{g}$ is square integrable) for the solution (155)
when we obtain

$ic_{0}^{\frac{3}{2}}|t+\gamma|^{-\frac{3}{2}}\Gamma=\frac{1}{2}|t+\gamma|^{-1}-\sqrt{c_{0}}k|t+\gamma|^{-\frac{1}{2}}\coth(2\sqrt{c_{0}}k\sqrt{-t-\gamma})<0$

For small $|t+\gamma|$, we have
$i\Gamma\simeq-\frac{5}{6}|t+\gamma|^{\frac{3}{2}}c_{0}^{-\frac{1}{2}}k^{2}$.
There can be a smooth limit $\gamma\rightarrow 0$ of quantum scalar field
theory when the metric passes from positive to negative signature. This
happens if $\Gamma$ in the WKB state (40) is determined by the classical wave
function solution (153) (for $t+\gamma>0$) and (155) (for $t+\gamma<0$).
$G_{t}$ is purely imaginary for a positive time, $G_{0}=0$ and $G_{t}$ becomes
a real positively definite function for a negative time. For the exponential
solutions, $\Re i\Gamma$ is also negative, as discussed at Equation (168). We
have obtained the dynamics of the fields $\Phi_{s}({\bf x})$ in all cases
(152)–(157). However, the limit $\gamma\rightarrow 0$ of the generate metric
exists only for solutions (153) and (155). The solution (153) (for positive
time) and its continuation to (155) (for negative time) are distinguished from
all solutions of the wave equation as the corresponding Gaussian wave function
being the WKB solution for positive time becomes a Gaussian normalizable wave
function for the negative time. This behavior resembles the one in the
elementary WKB approach when the wave function $\exp(\frac{i}{\hbar}\int
dx\sqrt{2(E-V)})$, before a potential barrier takes the form
$\exp(-\frac{1}{\hbar}\int dx\sqrt{2(V-E}))$ inside the barrier, suggesting a
tunneling process for the scalar field after crossing the classical barrier of
an inverted signature.

If $a(t)\simeq|t|^{\frac{1}{2}}$, then the quantum scalar field evolution is
unitary, and remains oscillatory, whereas for $a^{2}(t)\simeq t$, the
oscillatory behavior for positive time becomes a diffusion at negative time as
if the system encountered a barrier. In such a case, the evolution fails to be
unitary. If we stopped the evolution at certain space-like surface according
to the Hawking–Penrose singularity theorem [32], then unitarity would be
violated as well. If, in the Lagrangian (22) and in the Hamiltonian (24), we
replace $\sqrt{-g}$ by $\sqrt{|g|}$, then the resulting time evolution would
resemble the one of an upside-down oscillator (it would be still unitary after
an inversion of the signature).

Finally, we note that the correlation functions (157) are infinite if
$2k\sqrt{c_{0}}\sqrt{t+\gamma}=(n+\frac{1}{2})\pi$. In Equation (159), we
obtain a pole at $2\sqrt{c_{0}}\sqrt{t+\gamma}k=n\pi$, where $n$ is a natural
number. So the fields $\Phi_{s}(\Phi)$ are well-defined for small
$k\sqrt{t+\gamma}$. This difficulty already appears for a quantum mechanical
oscillator of frequency $k$ (see Appendix LABEL:secappA). It means that the
time evolution on the WKB states should be carefully extended from small
values of time. The problem is connected with caustic singularities in
semiclassical expansion [77].

##  11 Interaction with an Ultraviolet Cutoff

In this section, we discuss the Feynman–Kac formula for states of the WKB form
$\exp(\frac{i}{\hbar}\Phi\Gamma\Phi)\chi$, where $\Gamma$ is a real bilinear
form defined by a real solution of the wave equation. In models of Section 6,
the real solution $u$ (leading to a real $\Gamma$) is a real Bessel function
or the mode function in the expansion in spherical functions in Section 5. In
[36], we have shown in quantum mechanics and in QFT in the Minkowski space-
time that if the first (WKB) term on the rhs of Equation (44) is real, then
$G_{ss^{\prime}}$ is purely imaginary (as it is for the inverted oscillator).
In such a case for trigonometric or exponential interactions, the perturbation
series is absolutely convergent. In this section, we consider the free fields
of Sections 3–10 and potentials of the form

$V(\Phi)=\lambda\int_{B}d{\bf
x}\sqrt{|g|}d\mu(\alpha):\exp(i\alpha\Phi_{s}({\bf x})):,$ (172)

where $\mu$ is a complex measure, $B\subset R^{3}$ is a bounded domain, $:-:$
denotes the normal ordering, $\alpha$ can be real or purely imaginary, and
$\chi$ is of the form

$\chi(\Phi)=\int d\nu(\alpha_{0})\exp(i\alpha_{0}(f,\Phi))$ (173)

with $f\in L^{2}(d{\bf x})$. The exponential model ($\alpha$ imaginary)
appears as Starobinski model for inflation [78] and the trigonometric
interaction as the model of natural inflation [79].

We discuss also polynomial interactions of the form

$V(\Phi)=\lambda\int_{B}d{\bf x}\sqrt{|g|}:\Phi_{s}^{N}:({\bf x}),$ (174)

where $N=4n$ and $n$ is a natural number. In the latter case, we consider the
Feynman–Kac solution for the holomorphically extended initial wave function
$\psi(\sqrt{i}\Phi)$ (such states in the Feynman integral have been discussed
first in [34, 80]). Extensions of the wave functions (a complex scaling) are
studied in the theory of resonances [81, 82]. If in the expanding flat metric
(67), we put the scalar field in a spatial box of length $L$, then ${\bf k}$
is discrete ${\bf k}=\frac{2\pi{\bf n}}{L}$, with ${\bf
n}=(n_{1},n_{2},n_{3})$, where $n_{j}$ are integers. For a small time, the
Feynman formula will be well-defined if we restrict the range of ${\bf n}$.
First, we consider general formulas without specifying the number of ${\bf k}$
modes or $\Phi_{lm}$ modes (59). Then, we explain why a restriction to a
finite number of modes (or an ultraviolet cutoff) is necessary in the case of
the pseudoRiemannian metric.

The solution of the Schrödinger equation with the potential (172) (positive
time, a bounded region $B$) reads

$\begin{array}[]{l}\psi_{t}(\sqrt{i}\Phi)=\psi_{t}^{g}(\sqrt{i}\Phi)E\Big{[}\exp\Big{(}-\lambda\frac{i}{\hbar}\int_{B}d{\bf
x}\int_{0}^{t}dsa^{3}\exp(iN\frac{\pi}{4})\cr:\Big{(}u_{t-s}u_{t}^{-1}\Phi+\sqrt{\hbar}u_{t-s}\int_{0}^{s}(a^{\frac{3}{2}}u)^{-1}({t-\tau})dW_{\tau}\Big{)}^{N}:\Big{)}\chi(\sqrt{i}\Phi_{t}(\Phi))\Big{]}\end{array}$
(175)

In Equation (175), a necessary and sufficient condition for the stochastic
integral (82) to be well-defined is that the integral (112) for $G_{t}$ exists
(this is equivalent to $G_{ss^{\prime}}$ being finite). With the processes of
Section 10, this requirement can be achieved (because of the caustic poles
[77]) only for a small time if we have a finite number of modes or an
ultraviolet cutoff $\kappa$ ($k<\kappa$), so that $t\kappa$ is sufficiently
small. For a free field, the ultraviolet cutoff can be introduced by a
restriction of states $\chi$ in Equation (108) as Fourier transforms to
$\Lambda$, which have their support on $|{\bf k}|<\kappa$. Another way is to
introduce in the stochastic Equation (80), and in Equation (175), the
ultraviolet regularized Brownian motion $W_{s}^{\kappa}\rightarrow W_{s}$ with
$\kappa\rightarrow\infty$. The regularized Brownian motion is defined by the
covariance

$E[W^{\kappa}_{s}({\bf k})W^{\kappa}_{t}({\bf
k}^{\prime})]=min(s,t)\delta({\bf k}+{\bf k}^{\prime})\rho_{\kappa}(k),$ (176)

where $\rho$ is an ultraviolet cutoff restricting $k$ to $k\leq\kappa$
($\rho_{\kappa}(k)\rightarrow 1$ when $\kappa\rightarrow\infty$). If $N=4n$,
then the exponential in Equation (175) is bounded by 1. In such a case,

$\begin{array}[]{l}|\psi_{t}^{\kappa}(\sqrt{i}\Phi)|\leq
E\Big{[}|\psi(\sqrt{i}\Phi_{t}(\Phi))|\Big{]}\end{array}$ (177)

The rhs of Equation (177) is finite for a large class of functions (e.g., the
ones of Equation (173)). A renormalization of the polynomial interaction (175)
is required if the perturbation expansion in the coupling constant $\lambda$
is to be finite.

For the trigonometric interaction (172) an expansion of the exponential in
Equation (46) (inside $E[..]$) leads to integrals
$d\mu(\alpha_{j})d\nu(\alpha_{0})$ of functions of the form

$\begin{array}[]{l}\frac{\lambda^{n}}{n!}\int ds_{1}\ldots ds_{n}d{\bf
x}_{1}\ldots d{\bf x}_{n}E\Big{[}:\exp(i\alpha_{1}\Phi_{s_{1}}({\bf
x}_{1})):\ldots.:\exp(i\alpha_{n}\Phi_{s_{n}}({\bf
x}_{n})):\cr\times\exp(i\alpha_{0}\int d{\bf x}f({\bf x})\Phi_{t}({\bf
x}))\Big{]}.\end{array}$ (178)

The expectation value (178) is ($j\neq r$ because of the normal ordering)

$\begin{array}[]{l}\exp\Big{(}-\frac{1}{2}\sum_{j\neq
r}\alpha_{j}\alpha_{r}\int d{\bf k}\exp(i{\bf k}({\bf x}_{j}-{\bf
x}_{r}))G_{s_{j}s_{r}}(k)\Big{)}\end{array}$ (179)

times functions depending on the initial value $\Phi$

$\exp\Big{(}i\sum_{j}\alpha_{j}(u_{t-s_{j}}u_{t}^{-1}\Phi)({\bf
x}_{j})+i\alpha_{0}\int d{\bf x}(u_{0}u_{t}^{-1}\Phi)({\bf x})f({\bf
x})\Big{)}$ (180)

Most explicitly, the problem with the interaction in the Feynman–Kac Formula
(46) appears in the model of Section 10, where $G_{ss^{\prime}}$ has been
calculated exactly. For a finite number of modes and small $s_{j}<t$, the
covariance $G_{s_{j},s_{r}}(k)$ is well-defined (as seen in Equations (157)
and (180)) if the modes satisfy $2k\sqrt{c_{0}}\sqrt{t+\gamma}<\frac{\pi}{2}$.
The Formulas (175) and (178) do not extend to arbitrary time and arbitrarily
large $k$, because the Formula (157) gives an infinite $G_{ss^{\prime}}(k)$ if
$2k\sqrt{c_{0}}\sqrt{t+\gamma}=(n+\frac{1}{2})\pi$ (a pole at this value of
$k$). In Equation (159), we obtain a similar pole at
$2\sqrt{c_{0}}\sqrt{t+\gamma}k=n\pi$, where $n$ is a natural number. However,
the Lebesgue integral over $s_{j}$ and $\alpha_{j}$ in Equation (178) exists
for a finite number of modes if $G_{t}$ is purely imaginary, even if $k$ is
large so that the trigonometric functions have poles. This is a consequence of
the Lebesgue theorem saying that the Lebesgue integral of a bounded function
with singularities on a set of Lebesgue measure zero exists. We may hope that,
after an integration over $s_{j}$ and $\alpha_{j}$, we can go to the limit of
an infinite number of modes.

In de Sitter space in the massless case ($M=0$), we can obtain real solutions
$J_{\nu}$ and $Y_{\nu}$ of the wave equation. With a finite number of modes
for a small time, we can define the interaction (172) in the Feynman–Kac
formula in de Sitter space-time. The expression (179) will again be a pure
phase factor (bounded by 1). We are unable to calculate $G_{ss^{\prime}}$
explicitly in this case. However, we expect that an integral of oscillating
functions defining $G_{ss^{\prime}}$ will again show caustic singularities.
Such caustic poles appear already in the evolution kernel (the Mehler formula
for an oscillator) of free field theory. For a free field, even though the
Mehler kernel (see Appendix A) has the caustic singularities at
$t=\frac{\pi}{\omega}(n+\frac{1}{2})$, the calculation of expectation values
in the ground state gives correlation functions, which have an extension to an
arbitrary time (this may be a consequence of the fact that the ground state is
time-independent). We do not know whether such an extension of correlation
functions is possible in the models of Sections 8–10. In Sections 6 and 8, we
have discussed complex $u$, leading to an integrable $|\psi_{t}|^{2}$. Such a
wave function $u$ defines the free quantum field in de Sitter space-time and
in the universe expanding as $|t|^{\alpha}$. The correlation functions in such
states show no caustic poles. In particular, we can construct de Sitter
invariant correlation functions taking the Hankel function $H_{\nu}$ as the
solution of the wave Equation (see Equation (90)).

In the next section, we show that the inversion of the signature which removes
the caustics allows to define the Feynman integral without the ultraviolet
cutoff (or the restriction on the number of modes).

##  12 Inverted Metric without an Ultraviolet Cutoff

In [36], we have discussed the models (172)–(174) (see Appendixes C and E in
this paper) in the Minkowski space-time. We have obtained a convergent
perturbation series by an inversion of the signature of $(\nabla\Phi)^{2}$ in
the Lagrangian (3) (but in contradistinction to Euclidean quantum field theory
the time remains real). In the models of Sections 8–10, the inversion of the
signature means $k^{2}\rightarrow-k^{2}$ in Equation (68) or $k\rightarrow
ik$.

In Section 11, we have shown that if $\Gamma$ is real, then we have a well-
defined Feynman formula for a finite number of modes (or an ultraviolet
cutoff) and a small time $t$. In this section, we discuss the Feynman integral
for fields on a manifold with an inverted (Euclidean) metric in the
interpretation (27). Then, we have the Schrödinger equation for $t>0$, as well
as $t<0$ and $G_{ss^{\prime}}$ which is purely imaginary and without the
caustic poles, hence the model similar to that discussed in [36]. We obtain
such a Feynman integral if, in the integral over metrics (with the
interpretation of $\sqrt{-g}$ as $\sqrt{|g|}$ for an inverted metric), there
appear stationary points (as solutions of Einstein equations) with an inverted
signature. The stationary point may appear as the four-sphere (in addition to
the de Sitter space of Section 6), as the metric $a^{2}\simeq-t^{2}$ in the
coasting cosmology of Section 9, and in the background $a^{2}\simeq t$ of
Section 10, if the Feynman integral is considered for negative time. With the
inverted signature, there are no caustics in $G_{ss^{\prime}}$. If the term
$\sqrt{-g}$ in the Feynman integral (or in the Hamiltonian (24)) is replaced
by $\sqrt{|g|}$ (or if the inversion of signature is considered as a technical
tool on an intermediate stage of the construction of quantum fields), then
$G_{ss^{\prime}}$ will be purely imaginary. In such a case, we can apply the
Feynman formula of Section 11 to trigonometric and $\Phi^{4n}$ interactions
with an arbitrarily large $t$ and without an ultraviolet cutoff (owing to the
bound (177)). The ultraviolet limit exists, although the terms in the
perturbation expansion (in $\lambda$ in Equation (175) and in $\alpha$ in
Equation (172)) are divergent. Note that the Formula (175) is well-defined for
an arbitrarily large time with an ultraviolet cutoff, ensuring that the
functions in the exponential are bounded. Because of the bound (177), we may
apply the Lebesgue dominated convergence theorem to claim that the limit
$\kappa\rightarrow\infty$ with $\rho_{\kappa}(k)\rightarrow 1$ exists. The
interaction $\Phi^{4}$ deserves a detailed studies because of its role in the
standard model. A renormalization of the interaction is necessary if the
perturbation series in $\lambda$ is to be finite. The counterterms for
$\Phi^{4}$ are the same as in the conventional perturbation expansion because
the ultraviolet singularity of the stochastic field is the same as the one of
the quantum field.

We discuss the removal of regularization in more detail in the case of the
trigonometric interactions (172). Using the representation (172) and (173), we
calculate the expectation value over the Brownian motion $W_{s}^{\kappa}$ of
the n-th order term in the perturbation expansion. We obtain a perturbation
series of the form

$\begin{array}[]{l}\lambda^{n}\frac{1}{n!}\int\prod_{r}ds_{r}d{\bf
x}_{r}A(\Phi)\exp\Big{(}-\sum_{j\neq
r}\frac{1}{2}\alpha_{j}\alpha_{r}G^{\kappa}_{s_{j},s_{r}}({\bf x}_{j},{\bf
x}_{r})\Big{)},\end{array}$ (181)

where $A(\Phi)$ is a bounded function depending on the initial value $\Phi$ of
the field $\Phi_{s}$. The exponential factor in Equation (181) is a pure phase
as the covariance $G^{\kappa}_{s,s^{\prime}}$ is purely imaginary. As a
consequence, the perturbation series is absolutely convergent if $\int
d|\mu|<\infty$. In the model of Section 10 with the interpretation (27) of the
Hamiltonian for the negative time (when $a^{2}<0$), there will be no caustics
poles so that $G_{s_{j},s_{r}}({\bf x}_{j},{\bf x}_{r})$ (after the removal of
the ultraviolet cutoff) is well-defined outside the coinciding points, which
have the zero Lebesgue measure $ds_{1}d{\bf x}_{1}\ldots\ldots ds_{n}d{\bf
x}_{n}$. As a consequence of the Lebesgue theorem, the integral (181) exists
also after the removal of the ultraviolet cutoff. In the model of Section 9
with $\epsilon=-1$, the function $G_{ss^{\prime}}(k)$ is non-singular as a
function of k. The Fourier transform of $G_{ss^{\prime}}(k)$ in Equation (179)
is singular when $(s_{j},{\bf x}_{j})\rightarrow(s_{r},{\bf x}_{r})$, because
$G_{ss^{\prime}}(k)$ does not fall fast enough for large $k$. So,
$G_{ss^{\prime}}({\bf x}_{j}-{\bf x}_{r})$ is divergent at the coinciding
points $(s,{\bf x}_{j})\rightarrow(s^{\prime},{\bf x}_{r})$. This is the
reason (as discussed at the end of Section 2) that, in the Euclidean space,
the non-polynomial interactions cannot be defined. In the Minkowski space-
time, the two-point function is a complex distribution singular on the light-
cone, so that the Formula (181) would be untractable [71]. However, with the
real time and the inverted metric, the exponential in Equation (181) is a pure
phase bounded by 1. The integral over the singular points can be treated by
means of the Lebesgue lemma. According to the Lebesgue lemma, an integral of a
bounded function with singularities on a set of measure zero exists. When we
consider an ultraviolet regularized model (as in Section 11) then, applying
the Lebesgue dominated convergence theorem, we can conclude that, in the limit
$\kappa\rightarrow\infty$, the expression (178) is integrable and the
integrals are bounded by $|\lambda|^{n}|B|^{n}t^{n}\frac{1}{n!}$, because the
exponential (181) is a pure phase.

An analogous formula can be derived in the (Euclidean) de Sitter space in the
metric (56) and the interpretation (27) of the Hamiltonian. In such a case,
the process is generated by the Hamiltonian (61). Then, instead of the Formula
(181), we obtain (where $\omega_{j}$ are coordinates on $s^{3}$)

$\begin{array}[]{l}\int\prod_{j}ds_{j}d\omega_{j}\exp\Big{(}-\frac{1}{2}\sum_{j\neq
r}\alpha_{j}\alpha_{r}\int\sum_{lm}Y_{lm}(\omega_{r})Y_{lm}(\omega_{j})^{*}G^{lm}_{s_{j}s_{r}}\Big{)}\end{array}$
(182)

$G^{lm}_{s_{j}s_{r}}$ is purely imaginary and without caustic singularities.
The infinite sum in the exponential (182) is convergent to a purely imaginary
function singular at the coinciding points $(s_{j},\omega_{j})$ and
$(s_{r},\omega_{r})$. It has the same singularity as the Green function on
$S^{4}$. Hence, similarly as in Equation (181), we can prove the existence of
the integrals and their non-triviality using Lebesgue lemma on dominated
convergence.

In the metric (57), the analytic continuation of the metric to the Euclidean
(58) leads to a field theory on AdS. As discussed at the end of Section 5 by
means of the analytic continuation $k\rightarrow ik$, we obtain a real
solution $u(k)$ of the “wave equation” on the Euclidean anti-de Sitter space.
With the real $u$, the corresponding stochastic field has the
$G_{ss^{\prime}}$ correlation function which is purely imaginary. In such a
case, the expression (179) is integrable, defining the Euclidean quantum field
theory on AdS.

##  13 Summary

We have developed a functional Schrödinger description of the time evolution
of the scalar field in an external metric, which is a solution of Einstein
equations. We do not restrict ourselves to the solutions with the Lorentzian
signature, but also discuss solutions of Einstein equations with an Euclidean
signature. Such a metric can be relevant when averaging the scalar field
theory over all metrics in quantum gravity. In such a case, all saddle points
contributing to the path integral over the scalar field should be taken into
account. We work in the functional formulation of quantum field theory.
Solutions of the Schrödinger equation define a random field whose correlation
functions determine correlation functions of the quantum field. We consider
Gaussian solutions of the Schrödinger equation for free field theory. The
interaction is introduced by the Feynman–Kac formula. A proper choice of the
Gaussian solution facilitates a definition of an integrable Feynman–Kac
factor. We have shown that Gaussian solutions of the Schrödinger equation for
a free field theory are determined by the classical solutions of the wave
equation in an external metric. We studied in detail fields on de Sitter
space-time and on the flat expanding space-time. The stochastic field
correlation functions in a flat expanding space-time can be expressed by well-
known cylinder functions, whereas the ones in de Sitter space require infinite
series of Legendre functions. We have found a particular solution in a
radiation background, which is of the WKB form (a Gaussian phase factor) for a
positive time and a Gaussian square integrable function for a negative time
(one may consider it as composed from two solutions for positive and negative
time). This is an analog of the Gibbons–Hartle–Hawking solution in de Sitter
space-time, which is glued from a solution for a positive time and another one
for an imaginary time (positive and negative signature). We construct a
general solution of the Schrödinger equation as a perturbation of the Gaussian
(WKB) solution. We show how to calculate correlation functions of quantum
fields in terms of correlation functions of stochastic fields. Then, the
Feynman–Kac formula can be applied to construct solutions of the Schrödinger
equation with an interaction. We discuss polynomial and trigonometric
interactions. It is shown that the WKB solutions determined by real solutions
of the classical wave equation (with an inverted signature) are distinguished
from the point-of-view of the Feynman–Kac formula. The WKB solutions define
stochastic fields, which allow for the definition of the Feynman–Kac factor as
a bounded function. We briefly discussed the $\Phi^{4}$ interaction with a
positive and negative coupling, important for the standard model and models of
inflation. We pointed out that, with an inverted signature, an analytic
continuation and the standard renormalization, the Feynman–Kac factor can be
bounded and defined with finite renormalized perturbation series. This model
requires further investigation. We studied, in detail, the trigonometric
interaction showing that the perturbation series of the Schrödinger wave
function evolution can be expressed by pure phase factors. The phase factors