# Bogoliubov Transformation and Schrodinger Representation on Curved Space

Musfar Muhamed Kozhikkal<EMAIL_ADDRESS>Departament de Física
Quàntica i Atrofísica
Institut de Ciències del Cosmos
Universidad de Barcelona, Spain.  Arif Mohd<EMAIL_ADDRESS>Department of
Physics
Aligarh Muslim University, India.

###### Abstract

It is usually accepted that quantum dynamics described by Schrodinger equation
that determines the evolution of states from one Cauchy surface to another is
unitary. However, it has been known for some time that this expectation is not
borne out in the conventional setting in which one envisages the dynamics on a
fixed Hilbert space. Indeed it is not even true for linear quantum field
theory on Minkowski space if the chosen Cauchy surfaces are not preserved by
the flow of a timelike Killing vector. This issue was elegantly addressed and
resolved by Agullo and Ashtekar who showed that in a general setting quantum
dynamics in the Schrodinger picture does not take place in a fixed Hilbert
space. Instead, it takes place on a non-trivial bundle over time, the Hilbert
bundle, whose fibre at a given time is a Hilbert space at that time. In this
article, we postulate a Schrodinger equation that incorporates the effect of
change in vacuum during time evolution by including the Bogoliubov
transformation explicitly in the Schrodinger equation. More precisely, for a
linear (real) Klein-Gordon field on a globally hyperbolic spacetime we write
down a Schrodinger equation that propagates states between arbitrary chosen
Cauchy surfaces, thus describing the quantum dynamics on a Hilbert bundle. We
show that this dynamics is unitary if a specific tensor on the canonical phase
space satisfies the Hilbert-Schmidt condition. Generalized unitarity condition
of Agullo-Ashtekar follows quite naturally from our construction.

###### pacs:

###### Contents

  1. I Introduction
  2. II Scalar Field Theory: Classical
  3. III Scalar field theory: Quantum
    1. III.1 Geometric Quantization
    2. III.2 Bogoliubov transformation and Quadratic operators
    3. III.3 Unitarity of $\mathcal{A}$: The Hilbert-Schmidt condition
  4. IV Hamiltonian Evolution and The Schrodinger equation
  5. V Unitarity of Time Evolution
  6. VI Summary and Outlook
  7. A Normalization of the state

##  I Introduction

We do not know of any physical principle or algorithm that selects the vacuum
for quantum fields on globally hyperbolic curved spacetimes. The choice of the
vacuum is usually dictated by the symmetry of the background geometry,
positivity of energy, and by requiring the renormalizability of the stress
energy tensor. Indeed, even in Minkowski space there exist inequivalent
representations of the operator algebra (see, for e.g., ref. Wald (1995)). A
well-known example is the Rindler vs. Minkowski vacuum in flat space (the
former is reducible, however). The representation that is deemed physical is
the one compatible with the symmetries of the Minkowski space. Thus, for
quantum field theory on Minkowski space a unique vacuum, hence a unique
representation of the operator algebra, is picked out once the Poincare
invariance of the vacuum is imposed.

In the usual approach to quantum field theory on curved space (Ashtekar and
Magnon (1975); Ashtekar and Magnon-Ashtekar (1980)), one starts by quantizing
a classical phase space, which is usually taken to be the covariant phase
space. Classical theory also supplies the fields that are promoted to field
operators in the quantum theory. The choice of vacuum is encoded in the choice
of complex structure $J$ on the phase space. Complex structure is the new
mathematical structure needed to go from the classical theory to the quantum
theory. This leads to a one particle Hilbert space labeled by the points of
phase space and the total Hilbert space is constructed as a Fock space built
on top of the one particle Hilbert space. Each choice of $J$ thus corresponds
to a representation of the covariant operator algebra on a Hilbert space
labelled by $J$, $H_{J}$. This results in the covariant Heisenberg picture, in
which the field operators are time dependent and vacuum is a linear functional
on the (commutator) algebra of field operators. The dynamical information is
contained in the (representation of) operator algebra. The time dependent
vacuum correlation functions of field operators are the observables of the
theory.

In quantum mechanics, i.e., in the quantum description of finite number of
degrees of freedom, the unitary equivalence of Heisenberg and Schrodinger
representation is well known. The celebrated Stone-von Neumann theorem assures
that for finite number of particles the representation of canonical
commutation relations is essentially unique. The theorem however does not
apply to field theories due to the presence of infinite number of degrees of
freedom. One can thus ask if in the infinite dimensional setting, given a
quantum field theory in the Heisenberg representation, can a Schrodinger
representation be constructed so that one has the notion of a state that
evolves with time. In other words, the question is if in the field theory
setting can the dynamics be unitarily transferred from the operators to the
states in the Hilbert space. When the time evolution is along the timelike
Killing vector field of the background spacetime then this can be done. I.e,
when the spacetime is foliated in Cauchy slices orthogonal to the timelike
Killing symmetry of the spacetime then the Heisenberg and Schrodinger
representations are equivalent. That’s why one doesn’t ask this question in
most of the quantum field theory textbooks. However, many studies (see for
e.g., refs. Torre and Varadarajan (1999); Helfer (1999a, b, 1996); Cortez  _et
al._ (2013a, b); Gomar  _et al._ (2012); Cortez  _et al._ (2012); Gomez Vergel
and Villasenor (2008); Barbero G.  _et al._ (2008); Corichi  _et al._ (2007);
Cortez  _et al._ (2007); Corichi  _et al._ (2006); Torre (2002).) reported
that this is not true in general. More precisely, in different settings in
curved spacetimes, it was found in the mentioned studies that there is no
unitary Schrodinger dynamics between arbitrary Cauchy hypersurfaces (not
necessarily orthogonal to the time translation Killing field of the background
spacetime, or in the case that the background spacetime has no timelike
Killing vector, e.g., in a cosmological setting) in a fixed Hilbert space. In
particular, ref. Torre and Varadarajan (1999) showed that even in Minkowski
space, attempt to construct a unitary Schrodinger dynamics between arbitrary
Cauchy slices fails. But there is no problem in the covariant quantization
itself. It’s only that one can not transfer the dynamics from the operator
algebra to the states in a unitary fashion.

This perplexing state of affairs was addressed by Ashtekar and Agullo in an
elegant paper, Agullo and Ashtekar (2015). These authors realized that the
problem arises because of the insistence on describing the Schrodinger
dynamics on a fixed Hilbert space. The approach of covariant quantization can
be adapted to canonical quantization as well. In this case, one again needs to
choose a complex structure $J$, this time on the canonical phase space of the
theory. The old unitarity condition, the one that is presented in textbooks,
and the one that was shown to lead to the unitarity puzzle arises by asking if
there exists a unitary operator $U$ that evolves the state from initial time
slice $t_{i}$ to a final time slice $t_{f}$ such that the following is true
for any states $\ket{\Psi}$ and $\ket{\Psi^{\prime}}$ $\in H_{J}$

$\displaystyle\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}\hat{\mathcal{O}}(t_{i})\,U(t_{f},t_{i})\ket{\Psi}=\bra{\Psi^{\prime}}\hat{\mathcal{O}}(t_{f})\ket{\Psi},$
(1)

where the field operators on the left and right hand side $\hat{\mathcal{O}}$
are quantized with respect to $J$, i.e., they act on the fixed Hilbert space
$H_{J}$. On the right hand side the field operator is evolved to the final
time while the states are on the initial time. This is the Heisenberg
representation of quantum mechanics. On the left hand side the states are
evolved to the final time slice while the operator is on the initial time.
This is the Schrodinger representation of quantum mechanics. It turns out that
such a unitary operator $U$ exists if and only if $\left[J-EJE^{-1}\right]$ is
Hilbert-Schmidt111We will give a proof of the Hilbert-Schmidt condition in the
main text in sec. III.3., where $E$ is the map implementing time evolution
from $t_{i}$ to $t_{f}$ in the canonical phase space. In several examples on
curved space studied in refs. Torre and Varadarajan (1999); Helfer (1999a, b,
1996); Cortez  _et al._ (2013a, b); Gomar  _et al._ (2012); Cortez  _et al._
(2012); Gomez Vergel and Villasenor (2008); Barbero G.  _et al._ (2008);
Corichi  _et al._ (2007); Cortez  _et al._ (2007); Corichi  _et al._ (2006);
Torre (2002) this condition was shown to be violated. Even in simple
cosmological examples that were first studied in refs. Parker (1968, 1969,
1971), it was shown in ref. Agullo and Ashtekar (2015) that the Hilbert-
Schmidt condition is violated and hence such a unitary operator does not
exist.

The key insight of ref. Agullo and Ashtekar (2015) was that when the state
evolves from one Cauchy slice to the other, one needs to take care of the
evolution of the complex structure as well. Thus in the Schrodinger picture
representation, the quantum dynamics does not take place on a fixed Hilbert
space. Ref. Agullo and Ashtekar (2015) postulated a Generalized Unitarity
condition, eq. 2, which appropriately takes care of the evolution of the
complex structure. In this case, there is no kinematical unitary
identification between the Hilbert spaces on the initial and final Cauchy
slices, $H_{J(t_{i})}$ and $H_{J(t_{f})}$, respectively. Instead, there is a
dynamical unitary identification. Let’s see this in a bit more detail.

Let $\ket{\Psi}$ and $\ket{\Psi^{\prime}}$ be states $\in H_{J(t_{i})}$. Let
$U(t_{f},t_{i}):H_{J(t_{i})}\rightarrow H_{J(t_{f})}$, be the operator that
maps $H_{J(t_{i})}$ to $H_{J(t_{f})}$. Let $\hat{\mathcal{O}}$ be the operator
quantized in the representation $J(t_{i})$, thus it acts on states $\in
H_{J(t_{i})}$, and let
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{O}}}}$ be the
same operator quantized in the representation $J(t_{f})$, hence it acts on
$H_{J(t_{f})}$. The Generalized Unitarity condition of ref. Agullo and
Ashtekar (2015) is,

$\displaystyle\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathcal{O}}}}(t_{i})\,U(t_{f},t_{i})\ket{\Psi}=\bra{\Psi^{\prime}}\hat{\mathcal{O}}(t_{f})\ket{\Psi},$
(2)

where it should be noted that the hatted operator on the right hand side is
acting at final time $t_{f}$ but is quantized in the representation $J(t_{i})$
(thus acting on the Hilbert space at $t_{i}$, $H_{J(t_{i})}$), while the
double hatted operator on the left hand side is acting on the initial time
$t_{i}$ but is quantized in the representation $J(t_{f})$ (thus acting on the
Hilbert space at $t_{f}$, $H_{J(t_{f})}$). Therefore, the right hand side is
the Heisenberg representation in which operator is evolving with time while
the states and the representation space is fixed at the initial time. The left
hand side is the Schrodinger representation in which the states and thus the
representation space is evolving with time while the operator is acting at the
initial time. The departure of ref. Agullo and Ashtekar (2015) from the
previous studies, and the reason why eq. 2 is called the generalized unitarity
condition, is the fact that the Hilbert spaces $H_{J(t_{i})}$ and
$H_{J(t_{f})}$ are in general not unitarily equivalent, in the sense that
$\left[J(t_{f})-J(t_{i})\right]$ does not in general satisfy the Hilbert-
Schmidt condition. However, eq. 2 leads to a different unitarity condition,
namely, that $\left[J(t_{f})-EJ(t_{i})E^{-1}\right]$ should be Hilbert-
Schmidt, where $E$ is the map implementing time evolution from $t_{i}$ to
$t_{f}$ in the canonical phase space. In particular, if $J(t_{f})$ is the time
evolution of $J(t_{i})$ then we have that $J(t_{f})=EJ(t_{i})E^{-1}$ and the
Hilbert-Schmidt condition is trivially satisfied and the operator $U$ is
unitary. In the language of ref. Agullo and Ashtekar (2015), $U$ implements
unitarily the dynamical automorphism of the operator algebra.

The goal of this paper is to explicitly construct the operator $U$ of eq. 2.
Our work is complementary to that of ref. Agullo and Ashtekar (2015). While in
ref. Agullo and Ashtekar (2015) the generalized unitarity condition was
inspired by the classical picture and the existence of the covariant quantum
field theory and then deriving the Schrodinger picture representation, our
goal is to derive the generalized unitarity condition of ref. Agullo and
Ashtekar (2015) by postulating the explicit quantum dynamics in the
Schrodinger picture representation. We will use the ideas from geometric
quantization in order to quantize the canonical phase space. We will also find
it convenient to work in a fixed basis of the canonical phase space in a
discrete notation, thus treating it as if it were finite dimensional, and
keeping our eye on the infinite dimensional limit in which the issue of
unitarity arises.

The organization of this article is as follows. In sec. II we review the
construction of the canonical phase space for the real Klein-Gordon field
theory on a globally hyperbolic spacetime. One goal in this section is to show
that the symplectic form on the canonical phase space is translation
invariant, a fact that we will use in the geometric quantization of the
theory. Towards the end of this section we will also lay down our notation
that we will follow in the later sections. In sec. III we discuss the
quantization procedure following refs. Witten (1993); Axelrod  _et al._
(1991). In sec. III.1 we review the geometric quantization, construction of
the Hilbert space and quantization of linear functions. In sec. III.2 we
review the Bogoliubov transformation and the quantization of quadratic
functions. In sec. III.3 we show how the Hilbert-Schmidt condition arises as
the necessary condition for the unitary equivalence of two Hilbert spaces. In
sec. IV we present our Hamiltonian and the Schrodinger picture representation
of quantum dynamics of the Klein-Gordon field. In sec. V we make contact with
ref. Agullo and Ashtekar (2015) and derive their generalized unitarity
condition from our Schrodinger equation. We conclude with a brief summary and
outlook in sec. VI. In app. A we present the calculation for the normalization
of wavefunction that is used in sec. III.3.

##  II Scalar Field Theory: Classical

In this section we sketch the construction of the canonical phase space of the
classical free Klein-Gordon field theory. The purpose is to show that the
canonical Hamiltonian is a quadratic in fields and the symplectic structure is
translation invariant on the phase space. This is important for our
construction because we will be using this fact in the later sections.

Consider the free Klein-Gordon theory on a globally hyperbolic spacetime $M$
with a pseudo-Riemannian metric $g_{ab}$ described by the action

$\displaystyle
S=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\left(g^{ab}\nabla_{a}\Phi\nabla_{b}\Phi+m^{2}\Phi^{2}\right),$
(3)

where $g$ is the determinant of the metric, $g=det(g_{ab})$.

Let $\Sigma$ be a Cauchy surface. Given a one parameter family of spacelike
embeddings $\mathcal{I}_{t}$ of $\Sigma$ into $M$ we obtain a foliation of $M$
by a one parameter family of hypersurfaces
$\Sigma_{t}:=\mathcal{I}_{t}(\Sigma)$ specified by a time function
$t=constant$. Time evolution along the vector field
$t^{a}=\left(\frac{\partial}{\partial t}\right)^{a}$ can be decomposed in
terms of unit timelike normal to $\Sigma_{t}$, $n^{a}$ (where
$n_{a}=\nabla_{a}t$), the lapse function $N$, and the shift vector $N^{a}$. In
what follows we will put the shift to zero for simplicity. We then have the
following relations:

$\displaystyle g^{ab}$ $\displaystyle=h^{ab}-n^{a}n^{b},$ (4) $\displaystyle
t^{a}$ $\displaystyle=Nn^{a},$ (5)

where $h_{ab}$ is the metric induced on $\Sigma_{t}$. The Klein-Gordon action
can then be decomposed in a 3+1 form,

$\displaystyle
S=-\frac{1}{2}\int\mathrm{d}t\int_{\Sigma_{t}}\mathrm{d}^{3}xN\sqrt{h}\left(h^{ab}D_{a}\Phi
D_{b}\Phi-\frac{1}{N^{2}}({\cal L}_{t}\Phi)^{2}+m^{2}\Phi^{2}\right),$ (6)

where ${\cal L}_{t}\Phi$ is the Lie derivative of $\Phi$ along the time
evaluation vector field $t^{a}$, ${\cal L}_{t}\Phi=t^{a}\partial_{a}\Phi$, and
$D_{a}$ is the covariant derivative operator induced on the hypersurface
$\Sigma_{t}$ and is compatible with the induced metric $h_{ab}$.

Next, the canonical variables are defined as

$\displaystyle\phi(x)$ $\displaystyle=\Phi(t,x)|_{\Sigma_{t}}$ (7a)
$\displaystyle\pi(x)$ $\displaystyle=\frac{\delta S}{\delta{\cal
L}_{t}\Phi}=\frac{1}{N}{\cal L}_{t}\Phi(t,x)|_{\Sigma_{t}}.$ (7b)

The canonical Hamiltonian density is obtained by a Lagrange transform and we
get

$\displaystyle\mathcal{H}=\frac{1}{2}N\left(h^{ab}D_{a}\phi
D_{b}\phi+\pi^{2}+m^{2}\phi^{2}\right),$ (8)

This Hamiltonian generates the infinitesimal time evolution of the canonical
variables $\phi$ and $\pi$ via the Poisson brackets,

$\displaystyle{\cal L}_{t}\phi$ $\displaystyle=\\{\phi,\mathcal{H}\\}=N\pi,$
(9a) $\displaystyle{\cal L}_{t}\pi$
$\displaystyle=\\{\pi,\mathcal{H}\\}=h^{ab}D_{a}ND_{b}\phi+ND^{2}\phi-m^{2}\phi.$
(9b)

We thus arrive at the canonical phase space of the Klein Gordon theory. The
phase space $\Gamma$ is coordinatized by canonical variables $\phi(x)$ and
$\pi(x)$, i.e., a point on the phase space $\varphi$ is specified by its
coordinates $\varphi=(\phi,\pi)$. The poisson bracket is seen as the structure
on the phase space which is given by the symplectic structure, which is a non-
degenerate closed and exact two-form on the phase space,

$\displaystyle\omega(\delta_{1},\delta_{2}):=\int_{\Sigma_{t}}\mathrm{d}^{3}x\sqrt{h}\left(\frac{\delta_{1}}{\delta_{1}\phi}\frac{\delta_{2}}{\delta_{2}\pi}-\frac{\delta_{1}}{\delta_{1}\pi}\frac{\delta_{2}}{\delta_{2}\phi}\right).$
(10)

Owing to the linearity of $\Gamma$, the symplectic structure can be pulled
back from the tangent space of $\Gamma$ to $\Gamma$ thus giving $\Gamma$ the
structure of a symplectic vector space with the symplectic form, which we
again denote by $\omega$. Symplectic product between two phase space points
$\varphi_{1}=\left(\phi_{1},\pi_{1}\right)$ and
$\varphi_{2}=\left(\phi_{2},\pi_{2}\right)$ is given by

$\displaystyle\omega(\varphi_{1},\varphi_{2})=\int_{\Sigma_{t}}\mathrm{d}^{3}x\sqrt{h}\left(\phi_{1}(x)\pi_{2}(x)-\phi_{2}(x)\pi_{1}(x)\right).$
(11)

One reason to discuss this well-known classical setting is to emphasize that
the symplectic structure is translation invariant in the sense that its
components do not depend on the phase space point.

Finite time evolution, from $t_{i}$ to $t_{f}$ on the canonical phase space
can be obtained by integrating the Hamilton equations, eq. 9. Equivalently, it
can be written as a map222This is inverse of the map in ref. Agullo and
Ashtekar (2015).

$\displaystyle E_{t_{f}t_{i}}=\mathcal{I}_{t_{f}}^{-1}\mathcal{I}_{t_{i}}.$
(12)

The meaning of this equation is the following: for given functions $\phi(x)$
and $\pi(x)$ on $\Sigma$, first push them to $\Sigma_{t_{i}}$ using the map
$\mathcal{I_{t_{i}}}$. Treating this as initial data on $\Sigma_{t_{i}}$, use
the Klein Gordon equation $\left(\Box-m^{2}\right)\Phi(t,x)=0$ to find the
solution $\Phi(t,x)$ corresponding to the initial data $(\phi,\pi)$. Evaluate
the solution $\Phi(t,x)$ and its time derivative on the time slice
$\Sigma_{t_{f}}$ and push it back to $\Sigma$ using the map
$\mathcal{I}_{t_{f}}^{-1}$. This process gives the finite time evaluation of
the phase space point $\varphi$ at an initial time $t_{i}$ to a final time
$t_{f}$ as $E_{t_{f}t_{i}}\cdot\varphi$.

In what follows, we will work in a chosen basis for the canonical phase space.
We will find it convenient to use a discrete notation and denote this basis as
$\varphi^{i}$, where $i$ would in general stand for continuous indices like
$x$. The discussion of this section can thus be summarized as follows. In the
canonical description of the Klein-Gordon field we have a phase space $\Gamma$
whose coordinates are $\varphi^{i}$. Variation of any quantitiy with phase
space point will be denoted by $\partial_{i}$ which means
$\frac{\partial}{\partial\varphi^{i}}$. $\Gamma$ is equipped with a non-
degenerate closed and exact antisymmetric matrix, the canonical symplectic
structure, with components $\omega_{ij}$. The inverse matrix of $\omega_{ij}$
will be denoted by $\omega^{ij}$, such that
$\omega_{ij}\omega^{jk}=\delta^{k}_{i}$. Note that components of the
symplectic structure $\omega_{ij}$ are translation invariant in the sense that
they do not depend upon the phase space points $\varphi^{i}$’s.

Infinitesimal time evolution on the phase space is generated by the quadratic
Hamiltonian which we write in component form as
$\mathcal{H}=\frac{1}{2}H_{ij}\varphi^{i}\varphi^{j}$, where $H_{ij}$ is
symmetric in its indices. The vector field on $\Gamma$ corresponding to any
function $f$ on $\Gamma$ is given by $V_{f}=\omega^{-1}\mathrm{d}f$. For our
quadratic Hamiltonian this vector field has components
$V^{i}_{\mathcal{H}}=\omega^{ij}\partial_{j}\mathcal{H}=\omega^{ij}H_{jk}\varphi^{k}\equiv\left(\omega^{-1}H\right)^{i}_{~{}j}\varphi^{j}\equiv
T^{i}_{~{}j}\varphi^{j}$. Thus infinitesimal time evolution of phase space is
described by the matrix $T=\omega^{-1}H$. Finite time evolution on $\Gamma$ is
still given by the map $E_{t_{f}t_{i}}$ , i.e.,
$\varphi^{i}(t_{f})=E^{i}_{~{}j}\varphi^{j}(t_{i})$.

##  III Scalar field theory: Quantum

In order to quantize the canonical phase space in the previous section we
follow the ideas from geometric quantization (Witten (1993); Axelrod  _et al._
(1991)) which is in fact closely related to the general construction of
quantum field theory on curved spacetime in ref. Ashtekar and Magnon (1975)
(see also ref. Ashtekar and Magnon-Ashtekar (1980)) and used in ref. Agullo
and Ashtekar (2015). In both cases, one starts with the classical phase space
and introduces a complex structure on it that is compatible with the
symplectic structure in the sense discussed below.

Accessible reviews of geometric quantization can be found in refs. Nair
(2016); Woodhouse (1992). Here we will follow refs. Witten (1993); Axelrod
_et al._ (1991) to construct the quantum theory. In sec. III.1 we will discuss
the construction of the Hilbert space and the quantization of linear opeators.
In sec. III.2 we describe the Bogoliubov transformation and the quantization
of quadratic operators. In sec. III.3 we provide a proof of the Hilbert-
Schmidt condition.

###  III.1 Geometric Quantization

In geometric quantization one begins by first constructing a unitary line
bundle on the phase space whose curvature is $-i\omega_{ij}$. If the
symplectic structure is translation invariant then the covariant derivative on
the sections of this bundle can be taken to be

$\displaystyle\mathrm{D}_{i}=\partial_{i}+\frac{i}{2}\omega_{ij}\varphi^{j}.$
(13)

Square integrable sections of this line bundle constitute the pre-quantum
Hilbert space. Next one constructs a holomorphic structure on the pre-quantum
line bundle by introducing the complex structure $J$ on the phase space such
that: $J^{2}=-1$ and $J$ is translation invariant in the sense that its
components $J^{i}_{~{}j}$ do not depend upon the point on the phase space
where it is evaluated; $\omega_{ij}$ is compatible with $J$ in the sense that
$\omega_{ij}J^{i}_{~{}p}J^{j}_{~{}q}=\omega_{pq}$; $J$ is positive in the
sense that it defines a positive metric
$\text{g}_{ij}=\omega_{ik}J^{k}_{~{}j}$ on phase space.

The Hilbert space, i.e., the space of quantum states (wavefunctions), consists
of the square integrable sections of the pre-quantum Hilbert space which are
holomorphic with respect to $J$, i.e., they satisfy the following equation

$\displaystyle(1+iJ)^{k}_{~{}i}\mathrm{D}_{k}\Psi=0.$ (14)

Since this Hilbert space depends on the choice of complex structure $J$ it
will be denoted by $H_{J}$.

Following ref. Witten (1993) it would be very convenient to introduce some
notation that will streamline the calculations. Let us first introduce the
projectors that project on the holomorphic and anti-holomorphic sectors,

$\displaystyle\frac{1}{2}(1-iJ)^{i}_{~{}j}\,\,\,\,\,\,\,\,\,$
$\displaystyle\text{holomorphic},$ (15a)
$\displaystyle\frac{1}{2}(1+iJ)^{i}_{~{}j}\,\,\,\,\,\,\,\,\,$
$\displaystyle\text{anti-holomorphic}.$ (15b)

Projected components of vectors and one-forms will be represented by
over/under-lined indices as :

$\displaystyle
V^{\underaccent{\bar}{i}}=\frac{1}{2}(1-iJ)^{i}_{~{}j}V^{j}\,\,\,\,\,\,\,\,\,$
$\displaystyle\text{holomorphic},$ (16a) $\displaystyle
V^{\bar{i}}=\frac{1}{2}(1+iJ)^{i}_{~{}j}V^{j}\,\,\,\,\,\,\,\,\,$
$\displaystyle\text{anti-holomorphic},$ (16b) $\displaystyle
V_{\underaccent{\bar}{i}}=\frac{1}{2}(1-iJ)^{k}_{~{}i}V_{k}\,\,\,\,\,\,\,\,\,$
$\displaystyle\text{holomorphic},$ (16c) $\displaystyle
V_{\bar{i}}=\frac{1}{2}(1+iJ)^{k}_{~{}i}V_{k}\,\,\,\,\,\,\,\,\,$
$\displaystyle\text{anti-holomorphic}.$ (16d)

The non-zero components of the complex structure are given by
$J^{\underaccent{\bar}{i}}_{~{}\underaccent{\bar}{j}}=i\delta^{\underaccent{\bar}{i}}_{~{}\underaccent{\bar}{j}}$
and $J^{\bar{i}}_{~{}\bar{j}}=-i\delta^{\bar{i}}_{~{}\bar{j}}$. Furthermore,
due to the condition of compatibility between $\omega$ and $J$, the non-zero
components of the symplectic structure are
$\omega_{\underaccent{\bar}{i}\bar{j}}$ and
$\omega_{\bar{i}\underaccent{\bar}{j}}$.

In this notation, wavefunctions which lie in $H_{J}$ are holomorphic in the
sense that their anti-holomorphic components vanish,

$\displaystyle\mathrm{D}_{\bar{i}}\Psi=0.$ (17)

By using the definition of $\mathrm{D}$ in eq. 13, eq. 17 can be partially
solved to get that any holomorphic wavefunction $\Psi$ is of the form,

$\displaystyle\Psi=e^{-\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\varphi^{\bar{i}}\varphi^{\underaccent{\bar}{j}}}g(\varphi),$
(18)

where $g$ is a holomorphic function of coordinates in the ordinary sense,
i.e., $\partial_{\bar{i}}g=0,\forall i$. The wavefunction with $g=1$ is the
vacuum wavefunction in Hilbert space $H_{J}$,

$\displaystyle\Psi_{vac}=e^{-\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\varphi^{\bar{i}}\varphi^{\underaccent{\bar}{j}}},$
(19)

and is characterized by the fact that
$\hat{\varphi}^{\bar{i}}\Psi_{vac}=0,\forall i$, where $\hat{\varphi}$ is the
operator to be defined in eq. 23.

The inner-product on $H_{J}$ is given by

$\displaystyle\braket{\Psi_{1},\Psi_{2}}=\int\left(\prod_{i=1}^{2N}\,d\varphi^{i}\right)\sqrt{\det{\left(\frac{\omega}{2\pi}\right)}}\,\Psi_{1}^{\ast}\Psi_{2},$
(20)

where $2N$ is the dimension of phase space. In the infinite dimensional case,
which is the case of interest for us, we will take the limit $N\to\infty$ at
the end of our calculations.

Next we introduce some operators on $H_{J}$. In geometric quantization there
is a heuristic recipe to get operators from classical functions on phase
space. Given a function $f$ on phase space, the pre-quantum operator $\hat{f}$
is constructed as

$\displaystyle\hat{f}=-iV^{i}_{f}\mathrm{D}_{i}+f,$ (21)

where $V^{i}_{f}$ is the vector field generated by $f$ on the phase space,
i.e., $V^{i}_{f}=\omega^{ij}\partial_{j}f$. This recipe is often to be
augmented by the so called metaplectic corrections. For our purposes we will
not need that technology. We will be interested only in quantizing the phase
space functions which are at worst quadratic in phase space coordinates. Thus
the technique given in refs. Witten (1993); Axelrod  _et al._ (1991) would be
enough for our purpose. To begin with, we define the operators corresponding
to linear functions on the phase space. The phase space coordinates
$\varphi^{i}$ form a basis for these functions. Following the geometric
quantization recipe, we get the corresponding operators as

$\displaystyle\hat{\varphi}^{i}=i\omega^{ij}\mathrm{D}_{j}+\varphi^{i},$ (22)

where we evaluated the vector field corresponding to $\varphi^{i}$ as
$V^{j}_{\varphi^{i}}=\omega^{jk}\partial_{k}\varphi^{i}=\omega^{ji}$. In
particular, for holomorphic and anti-holomorphic components, using the fact
that the wavefunctions are holomorphic (see eq. 17) we get,

$\displaystyle\hat{\varphi}^{\underaccent{\bar}{i}}$
$\displaystyle=\varphi^{\underaccent{\bar}{i}},$ (23a)
$\displaystyle\hat{\varphi}^{\bar{i}}$
$\displaystyle=i\omega^{\bar{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{j}}+\varphi^{\bar{i}}.$
(23b)

Before we discuss the quantization prescription for the quadratic operators we
need to understand how to take care of change in the complex structure. The
reason is that we will ultimately be interested in writing the Schrodinger
equation, for which we need the Hamiltonian. However, in a general dynamical
problem the complex structure will change with time. Since the Hamiltonian is
the infinitesimal generator of time evaluation, the finite time evolution
should take us from the initial Hilbert space $H_{J(t_{i})}$ to the final
Hilbert space $H_{J(t_{f})}$. We therefore turn to Bogoliubov transformation
that implements the change of $J$ in quantum operators. This was constructed
in Refs. Axelrod  _et al._ (1991); Witten (1993) and we follow one of the
constructions given in these references.

###  III.2 Bogoliubov transformation and Quadratic operators

Let us consider a quadratic Hamiltonian
$\mathcal{h}=\frac{1}{2}h_{ij}\varphi^{i}\varphi^{j}$, where $h_{ij}=h_{ji}$.
Vector field generated by $\mathcal{h}$ on the phase space is

$\displaystyle V_{\mathcal{h}}^{i}$
$\displaystyle=\omega^{ij}\partial_{j}\mathcal{h}$
$\displaystyle=\omega^{ij}h_{jk}\varphi^{k}$
$\displaystyle\equiv\left(\omega^{-1}h\right)^{i}_{~{}k}\varphi^{k}.$ (24)

Canonical transformation generated by $\mathcal{h}$ is the flow of
$V_{\mathcal{h}}$ on the phase space obtained by the Lie-drag due to
$V_{\mathcal{h}}$ ,

$\displaystyle\delta_{\mathcal{h}}\phi^{i}$ $\displaystyle={\cal
L}_{V_{\mathcal{h}}}\varphi^{i}$
$\displaystyle=V_{\mathcal{h}}^{j}\partial_{j}\varphi^{i}$
$\displaystyle\equiv\left(\omega^{-1}h\right)^{i}_{~{}j}\varphi^{j}.$ (25)

Change in the complex structure $J$ under this canonical transformation can be
obtained by computing the Lie-derivative of $J$ along $V_{\mathcal{h}}$. For
translation invariant complex structures (i.e., $\partial_{k}J^{i}_{~{}j}=0$)
we get

$\displaystyle\delta_{\mathcal{h}}J^{i}_{~{}j}$ $\displaystyle={\cal
L}_{V_{\mathcal{h}}}J^{i}_{~{}j}$
$\displaystyle=\left[J,\omega^{-1}h\right]^{i}_{~{}j}$ (26)

Now, choosing $\omega^{-1}h=-\frac{1}{2}\left(J\delta J\right)$, and (since
$J^{2}=-1$) using that $\delta J^{2}=0=\left(J\delta J+\delta JJ\right)$, we
get

$\displaystyle\delta_{\mathcal{h}}J^{i}_{~{}j}=\delta J^{i}_{~{}j}.$ (27)

Thus we see that any change in the complex structure $\delta J$ can be
obtained as a Hamiltonian flow on the phase space generated by the quadratic
Hamiltonian $\mathcal{h}=-\frac{1}{4}\left(\omega J\delta
J\right)_{ij}\varphi^{i}\varphi^{j}$. The vector field generated by
$\mathcal{h}$ is $V_{\mathcal{h}}=-\frac{1}{2}\left(J\delta
J\right)^{i}_{~{}j}\varphi^{j}$.

Follow the geometric quantization recipe for the pre-quantum operator
corresponding to the classical function $\mathcal{h}$ we get the pre-quantum
operator $Pr(\mathcal{h})$,

$\displaystyle Pr(\mathcal{h})$
$\displaystyle=-iV_{\mathcal{h}}^{i}\mathrm{D}_{i}+\mathcal{h}$
$\displaystyle=\frac{i}{2}\left(J\delta
J\right)^{i}_{~{}j}\varphi^{j}\mathrm{D}_{i}-\frac{1}{4}\left(\omega J\delta
J\right)_{ij}\varphi^{i}\varphi^{j}.$ (28)

Note that the pre-quantum operator is a first-order differential operator.
Following ref. Axelrod  _et al._ (1991) (see ref. Witten (1993) for another
approach) let’s write another operator corresponding to $\mathcal{h}$, denoted
by $\hat{\mathcal{h}}$, which is obtained by simply putting hats on
$\varphi$’s,

$\displaystyle\hat{\mathcal{h}}$
$\displaystyle=\frac{1}{2}h_{ij}\hat{\varphi}^{i}\hat{\varphi}^{j}$ (29)
$\displaystyle=\frac{1}{2}h_{ij}\left(i\omega^{im}\mathrm{D}_{m}+\varphi^{i}\right)\left(i\omega^{jn}\mathrm{D}_{n}+\varphi^{j}\right).$
(30)

where we don’t need to symmetrize since $h_{ij}$ is already symmetric. Note
that $\hat{\mathcal{h}}$ does not depend upon the complex structure $J$.
Expanding the right hand side we get that

$\displaystyle\hat{\mathcal{h}}=Pr(\mathcal{h})-\frac{1}{4}\left(J\delta
J\omega^{-1}\right)^{ij}\mathrm{D}_{i}\mathrm{D}_{j}.$ (31)

The meaning of this equation is as follows. The operator $\hat{\mathcal{h}}$
on the left hand side does not depend on $J$. The first term on the right hand
side, $Pr(\mathcal{h})$, implements the Bogoliubov transformation. Therefore,
the second term on the right hand side is interpreted as canceling the change
of complex structure. Since $\mathrm{D}$’s act on wavefunctions $\Psi$ which
are holomorphic with respect to $J$, and using that
$J^{\underaccent{\bar}{i}}_{~{}\underaccent{\bar}{j}}=i\delta^{\underaccent{\bar}{i}}_{~{}\underaccent{\bar}{j}}$,
the second term in eq.31 can be written as

$\displaystyle\mathcal{A}:=\frac{1}{4}\left(\delta
J\omega^{-1}\right)^{\underaccent{\bar}{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{j}},$
(32)

where the symbol $\mathcal{A}$ comes from the fact that $\mathcal{A}$ is the
connection on a bundle. Indeed, in the approach of ref. Axelrod  _et al._
(1991), one constructs a bundle whose base space is the space of $J$’s and
whose fibre at $J$ is the Hilbert space $H_{J}$. Then $\mathcal{A}$ denotes
the connection that is to be used for transporting a state from the fibre
$H_{J}$ at $J$ to the fibre $H_{J^{\prime}}$ at $J^{\prime}$. For finite
dimensional systems this connection is unitary, in the sense that the states
are mapped by this transport in a unitary fashion. In the infinite dimensional
case though this connection will generically be non-unitary. However, we will
see that this is not a problem. The map that would take us from the Hilbert
space at initial time $t_{i}$ to a final time $t_{f}$ will in fact be unitary,
thus demonstrating the implementation of “dynamical automorphism” of operator
algebra as a unitary transformation between the two Hilbert spaces as
described in ref. Agullo and Ashtekar (2015).

This leads us to the prescription of ref. Axelrod  _et al._ (1991) for the
quantization of classical functions quadratic in phase space coordinates. For
such a function $f$, the quantum operator $\hat{f}$ is given by

$\displaystyle\hat{f}$
$\displaystyle=Pr(f)+\frac{i}{4}\left(\delta_{f}J\omega^{-1}\right)^{\underaccent{\bar}{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{j}},$
(33)

where $Pr(f)=-iV_{f}^{i}\mathrm{D}_{i}+f$, and $\delta_{f}J$ is the change in
complex structure along the flow of the vector field $V_{f}$ on the phase
space, i.e., $\delta J={\cal L}_{V_{f}}J$. It can be checked that this
quantization scheme respects the Poisson bracket structure,

$\displaystyle[\hat{f},\hat{\varphi}^{k}]=-i\widehat{\left\\{f,\phi^{k}\right\\}}.$
(34)

###  III.3 Unitarity of $\mathcal{A}$: The Hilbert-Schmidt condition

Consider the phase space $\Gamma$ for a free scalar field theory. Let’s
construct two Hilbert spaces $H_{1}$ and ${H}_{2}$ corresponding to two
complex structures $J_{1}$ and $J_{2}$, respectively, on $\Gamma$. Then
$H_{1}$ and $H_{2}$ are unitarily equivalent iff the following condition is
satisfied

$\displaystyle\mathrm{tr}\,\,|J_{1}-J_{2}|^{2}<\infty,$ (35)

where the trace is over the phase space333which is equivalently the trace over
one-particle Hilbert corresponding to either $H_{1}$ or $H_{2}$ after Cauchy
completion.. The inequality in eq. 35 is called the Hilbert-Schmidt condition.
Here we provide a physicist’s derivation of the Hilbert-Schmidt condition
following ref. Ashtekar and Magnon-Ashtekar (1980).

We could use the connection to propagate the vacuum wavefunction (see, eq. 19)
with respect to $J_{1}$ and checking if the resulting wavefunction $\in
H_{J_{2}}$, i.e., if it is normalizable in $H_{J_{2}}$. The propagation is
implemented by the exponentiated version of $\mathcal{A}$, which is defined as
$\mathcal{U}=\mathcal{P}e^{\int\mathcal{A}}$ (path integrated exponential)
that maps the state from $H_{J_{1}}$ to $H_{J_{2}}$. However, it is easier to
follow another route as in ref. Ashtekar and Magnon-Ashtekar (1980). To this
end, we first compute that commutator of the connection with a field operator
$\hat{\varphi}^{i}$ and we get,

$\displaystyle\left[\mathcal{A},\hat{\varphi}^{i}\right]=0.$ (36)

This means that $U$ is such that

$\displaystyle
U^{-1}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{i}U=\hat{\varphi}^{i},$
(37)

where, since $U:H_{J_{1}}\rightarrow H_{J_{2}}$, we have that the double
hatted operator on the left hand side is quantized with respect to $J_{2}$ and
the one on the right hand side is quantized with respect to $J_{1}$. In
particular,

$\displaystyle
U^{-1}\left(1+iJ_{1}\right)\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}U=\left(1+iJ_{1}\right)\hat{\varphi},$
(38)

where we have omitted the indices for brevity. Operating the right hand side
on the vacuum in $H_{J_{1}}$, $\Psi_{vac}$ (see eq. 19), we get zero. Let
$\psi=U\Psi_{vac}$. Then we have that

$\displaystyle\left(1+iJ_{1}\right)^{i}_{~{}j}\hat{\varphi}^{j}\Psi_{vac}$
$\displaystyle=0,$
$\displaystyle\implies\left[\left(1+iJ_{1}\right)^{i}_{~{}\bar{j}}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{j}}+\left(1+iJ_{1}\right)^{i}_{~{}\underaccent{\bar}{j}}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{j}}\right]\psi=0,$
$\displaystyle\implies\left[\left(J_{2}+J_{1}\right)^{i}_{~{}\bar{j}}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{j}}-\left(J_{2}-J_{1}\right)^{i}_{~{}\underaccent{\bar}{j}}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{j}}\right]\psi=0,$
$\displaystyle\implies\left[\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{i}}-\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{j}}\right]\psi=0,$
(39)

where $\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\phi}}}$ again
signifies that the indices of the operator are projected with respect to
$J_{2}$, and $\chi$ is the operator $\chi:=(J_{2}+J_{1})^{-1}(J_{2}-J_{1})$ .
Next we substitute the operator definitions (see eq. 23) to get the
differential equation

$\displaystyle\left[i\omega^{\bar{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{j}}+\varphi^{\bar{i}}-\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}{\varphi}^{\underaccent{\bar}{j}}\right]\psi=0.$
(40)

This equation can be simplified by first expressing $\psi$ as
$\psi=e^{-\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\varphi^{\bar{i}}\varphi^{\underaccent{\bar}{j}}}g(\varphi)$,
for some holomorphic (with respect to $J_{2}$) function $g$, (see eq. 18),
which gives a differential equation for $g$ as

$\displaystyle\left[i\omega^{\bar{i}\underaccent{\bar}{j}}\partial_{\underaccent{\bar}{j}}-\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}{\varphi}^{\underaccent{\bar}{j}}\right]g=0,$
(41)

which is easily solved to get

$\displaystyle
g(\varphi)=e^{\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\chi^{\bar{i}}_{~{}\underaccent{\bar}{k}}\varphi^{\underaccent{\bar}{k}}\varphi^{\underaccent{\bar}{j}}}$
(42)

Therefore the vacuum $\Psi_{vac}\in H_{J_{1}}$ propagated to $H_{J_{2}}$
yields the state $\psi=U\Psi_{vac}$,

$\displaystyle\psi=\mathscr{N}\,e^{-\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\varphi^{\bar{i}}\varphi^{\underaccent{\bar}{j}}}e^{\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\chi^{\bar{i}}_{~{}\underaccent{\bar}{k}}\varphi^{\underaccent{\bar}{k}}\varphi^{\underaccent{\bar}{j}}},$
(43)

where $\mathscr{N}$ is yet-to-be-determined normalization constant. $\psi$
would be a legitimate state and would lie in $H_{J_{2}}$ if $\mathscr{N}$ is
finite. Calculation of the normalization factor is given in the appendix A,
where we show that the condition for the finiteness of $\mathscr{N}$ is that
$\mathrm{tr}\,\chi^{2}:=\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}\chi^{\underaccent{\bar}{j}}_{~{}\bar{i}}$
should be finite, which translates to the Hilbert-Schmidt condition

$\displaystyle\left(J_{2}-J_{1}\right)^{\bar{i}}_{~{}\underaccent{\bar}{k}}\left(J_{2}-J_{1}\right)^{\underaccent{\bar}{k}}_{~{}\bar{i}}<\infty.$
(44)

##  IV Hamiltonian Evolution and The Schrodinger equation

Let’s now consider a linear system whose time evaluation is given by a
Hamiltonian quadratic in coordinates, as it is the case for us (see eq. 8),

$\displaystyle\mathcal{H}=\frac{1}{2}H_{ij}\varphi^{i}\varphi^{j}.$ (45)

From the quantization prescription in eq. 33 for such a function we have the
corresponding operator,

$\displaystyle\hat{\mathcal{H}}=Pr(H)+\frac{i}{4}\left(\delta_{H}J\omega^{-1}\right)^{\underaccent{\bar}{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{j}},$
(46)

where $\delta_{H}J$ is the change in complex structure due to flow generated
by $H$ on phase space. While one can choose the complex structure at each time
independently, we will consider the natural time evolution of $J$. This means
that $\delta_{H}J={\cal L}_{V_{H}}J=\left[J,T\right]$, where, recall from the
last paragraph of sec. II, $T$ is the matrix describing the infinitesimal time
evolution of the phase space points
$(\dot{\varphi}^{i}=T^{i}_{~{}j}\varphi^{j})$. The reason that we keep only
the natural time evolution of the initial $J$ in our Hamiltonian is that once
we have the complete picture of time evolution of wavefunctions in this case,
then we can simply do a Bogoliubov transformation to any other complex
structure that we decide to choose on the final time slice. More discussion
about this appears at the end of sec. V.

It can be checked that with this prescription the quantization respects the
Poisson bracket,

$\displaystyle\left[\hat{\mathcal{H}},\hat{\varphi}^{k}\right]=-i\widehat{\left\\{\mathcal{H},\varphi^{k}\right\\}}.$
(47)

Importantly, our Hamiltonian respects the holomorphicity of wavefunctions.
This is so because we can check that

$\displaystyle\left[\hat{\mathcal{H}},\mathrm{D}_{\bar{i}}\right]=0.$ (48)

To see that this condition is necessary for preserving the holomorphicity of
wavefunctions let us see the implication of
$\delta\left(\mathrm{D}_{\bar{i}}\Psi\right)=0$,

$\displaystyle\delta\left(\mathrm{D}_{\bar{i}}\Psi\right)$
$\displaystyle=\delta\mathrm{D}_{\bar{i}}\cdot\Psi+\mathrm{D}_{\bar{i}}\,\left(\delta\Psi\right)$
$\displaystyle=i\left[\hat{\mathcal{H}},\mathrm{D}_{\bar{i}}\right]\Psi+\mathrm{D}_{\bar{i}}\left(-i\hat{\mathcal{H}}\Psi\right)$
$\displaystyle=i\left[\hat{\mathcal{H}},\mathrm{D}_{\bar{i}}\right]\Psi-i\left[\mathrm{D}_{\bar{i}},\hat{\mathcal{H}}\right]\Psi$
$\displaystyle=2i\left[\hat{\mathcal{H}},\mathrm{D}_{\bar{i}}\right]\Psi,$
(49)

where in going to the third equality from the second we used the
holomorphicity of $\Psi$. Therefore,
$\delta\left(\mathrm{D}_{\bar{i}}\Psi\right)=0$ $\implies$
$\left[\hat{\mathcal{H}},\mathrm{D}_{\bar{i}}\right]=0$. Thus, eq. 48 says
that our Hamiltonian keeps the holomorphic wavefunctions holomorphic.

Our proposal for the Schrodinger equation, i.e., the equation describing the
time evolution of the wavefunction is then

$\displaystyle i\frac{\partial\psi}{\partial t}=\hat{\mathcal{H}}\psi,$ (50)

where $\hat{\mathcal{H}}$ is as given in eq. 46 and includes the connection
term. Eq. 50 is the key equation of this paper.

From our Schrodinger equation in eq. 50 we have the following finite time
evolution of a quantum state,

$\displaystyle\Psi(t_{f})=U(t_{f},t_{i})\Psi(t_{i}),$ (51)

where,

$\displaystyle
U(t_{f},t_{i})=\mathcal{T}\left(e^{-i\int_{t_{i}}^{t_{f}}\mathrm{d}t\,\hat{\mathcal{H}}}\right),$
(52)

where $\mathcal{T}$ is the symbol for time ordering dictated by our foliation
of spacetime. The commutation relation in eq. 48 says that our Hamiltonian at
time $t$, which depends upon the complex structure $J$ at time $t$, preserves
the holomorphicity of wavefunctions at time $t$. This implies that the
exponentiation yielding the operator for finite time evolution
$U(t_{f},t_{i})$ maps the wavefunctions at time $t_{i}$ that are holomorphic
with respect to $J(t_{i})$ to the wavefunctions at time $t_{f}$ that are
holomorphic with respect to $J(t_{f})$. Thus $U(t_{f},t_{i})$ maps the Hilbert
space at time $t_{i}$, $H_{J(t_{i})}$ to the Hilbert space at time $t_{f}$,
$H_{J(t_{f})}$,

$\displaystyle U(t_{f},t_{i}):H_{J{(t_{i})}}\rightarrow H_{J{(t_{f})}}.$ (53)

The question now is, under what condition is $U$ a unitary operator? In order
to answer this, we could just propagate the vacuum state $\Psi_{vac}$ in
$H_{J_{t_{i}}}$ using $U$ and check for the finiteness of the norm of the
evolved state $U(t_{f},t_{i})\Psi_{vac}$ to see if it lies in $H_{J(t_{f})}$.
But it is easier to follow along the lines of analysis in sec. III.3. In the
next section we will first derive the Generalized Unitarity condition of ref.
Agullo and Ashtekar (2015) and then use it to prove the unitarity of $U$.

##  V Unitarity of Time Evolution

Let us begin by collecting at one place our expression of the quantized
Hamiltonian operator,

$\displaystyle\hat{\mathcal{H}}=Pr{(H)}+\frac{i}{4}\left(\delta_{H}J\omega^{-1}\right)^{\underaccent{\bar}{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{j}},$
(54)

where,

$\displaystyle
Pr(H)=-iV_{H}^{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{i}}+H$
$\displaystyle=-iT^{\underaccent{\bar}{i}}_{~{}k}\varphi^{k}\mathrm{D}_{\underaccent{\bar}{i}}+\frac{1}{2}(\omega
T)_{ij}\varphi^{i}\varphi^{j},$ (55)
$\displaystyle\frac{i}{4}\left(\delta_{H}J\omega^{-1}\right)^{\underaccent{\bar}{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{j}}$
$\displaystyle=\frac{1}{2}(T\omega^{-1})^{\underaccent{\bar}{i}\underaccent{\bar}{j}}\mathrm{D}_{\underaccent{\bar}{i}}\mathrm{D}_{\underaccent{\bar}{j}},$
(56)

where we have used that $\delta_{H}J={\cal L}_{V_{H}}J=[J,T]$. Furthermore,
the following commutation relations follow,

$\displaystyle
i\left[\hat{\mathcal{H}},\hat{\varphi}^{\underaccent{\bar}{i}}\right]$
$\displaystyle=T^{\underaccent{\bar}{i}}_{~{}k}\hat{\varphi}^{k},$ (57a)
$\displaystyle i\left[\hat{\mathcal{H}},\hat{\varphi}^{\bar{i}}\right]$
$\displaystyle=T^{\bar{i}}_{~{}{k}}\hat{\varphi}^{{k}}.$ (57b)

Consider an initial state $\Psi\in H_{J(t_{i})}$ and its time evolved
image444There is a slight abuse of notation here, for we don’t know yet if the
evolved state is $\in H_{J(t_{f})}$. $U(t_{f},t_{i})\Psi\in H_{J(t_{f})}$. Now
an operator that acts on the time evolved state is
${\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{k}}}$.
Notice that here $\underaccent{\bar}{k}$ means that the index $k$ is
holomorphically projected with respect to $J(t_{f})$, i.e, the operator is
quantized with respect to $J(t_{f})$ since it acts on the Hilbert space
$H_{J(t_{f})}$. The latter fact is emphasized by the double-hatted notation.
Similarly,
${\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{k}}}$
also acts on $H_{J(t_{f})}$ . Note that we want to interpret these as
Schrodinger picture operators so we are inserting them still at the initial
time $t_{i}$, but they act on the Hilbert space at $t_{f}$.

For any states $\ket{\Psi^{\prime}}$ and $\ket{\Psi}\in H_{J(t_{i})}$,
infinitesimal form of the operator insertion in
$\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{i}}}\,U(t_{f},t_{i})\ket{\Psi}$
and
$\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{i}}}\,U(t_{f},t_{i})\ket{\Psi}$,
in the limit that $t_{f}\rightarrow t_{i}$ can be calculated to be,

$\displaystyle
i\left[\hat{\mathcal{H}},\hat{\varphi}^{\underaccent{\bar}{i}}\right]-\frac{i}{2}\delta_{H}J^{i}_{~{}j}\hat{\varphi}^{j}$
$\displaystyle=T^{i}_{~{}\underaccent{\bar}{j}}\hat{\varphi}^{\underaccent{\bar}{j}},$
(58a) $\displaystyle
i\left[\hat{\mathcal{H}},\hat{\varphi}^{\bar{i}}\right]+\frac{i}{2}\delta_{H}J^{i}_{~{}j}\hat{\varphi}^{j}$
$\displaystyle=T^{i}_{~{}\bar{j}}\hat{\varphi}^{\bar{j}},$ (58b)

respectively. From these infinitesimal forms we deduce the corresponding
finite forms,

$\displaystyle\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{i}}}\,U(t_{f},t_{i})\ket{\Psi}=\bra{\Psi^{\prime}}E^{i}_{~{}j}\left(\frac{1-iJ(t_{f})}{2}\right)^{j}_{~{}k}\hat{\varphi}^{k}\ket{\Psi},$
(59a)
$\displaystyle\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{i}}}\,U(t_{f},t_{i})\ket{\Psi}=\bra{\Psi^{\prime}}E^{i}_{~{}j}\left(\frac{1+iJ(t_{f})}{2}\right)^{j}_{~{}k}\hat{\varphi}^{k}\ket{\Psi},$
(59b)

respectively. Here $E^{i}_{~{}j}$ is the classical time evaluation on the
canonical phase space $\Gamma$,
$\left(\varphi^{i}(t_{f})=E^{i}_{~{}j}\varphi^{j}(t_{i})\right)$ , and the
operator on the right hand side is quantized with respect to $J(t_{i})$ since
it acts on the states in the initial Hilbert space $H_{J(t_{i})}$. Adding
these two equations we get,

$\displaystyle\bra{\Psi^{\prime}}U(t_{f},t_{i})^{-1}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{{i}}}\,U(t_{f},t_{i})\ket{\Psi}=\bra{\Psi^{\prime}}E^{i}_{~{}j}\hat{\varphi}^{j}\ket{\Psi},$
(60)

where again the operator on the right hand side is quantized with respect to
$J(t_{i})$ while the one on the left hand side is quantized with respect to
$J(t_{f})$. Eq. 60 is precisely the Generalized Unitarity condition (eq. 2)
proposed in ref. Agullo and Ashtekar (2015). If the operator $U$ is unitary
then eq. 60 establishes the equivalence of the Heisenberg and Schrodinger
picture in quantum field theory for arbitrary foliation of the background
spacetime.

We turn to the analysis of unitarity of $U$ now. The strategy would be the
same as in sec. III.3. We project the operator on the right hand side to its
anti-holomorphic components and act on the vacuum state in the initial Hilbert
space. We have the following operator relation from eq. 60

$\displaystyle
U(t_{f},t_{i})^{-1}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{{i}}}\,U(t_{f},t_{i})=E^{i}_{~{}j}\hat{\varphi}^{j}.$
(61)

To get the anti-holomorphic components on the right hand side we perform the
following manipulations (suppressing the indices for a little bit),

$\displaystyle E\left(1+iJ(t_{i})\right)\hat{\varphi}$
$\displaystyle=\left(1+iEJ(t_{i})E^{-1}\right)E\hat{\varphi}$
$\displaystyle=U(t_{f},t_{i})^{-1}\left(1+iEJ(t_{i})E^{-1}\right){\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}}\,U(t_{f},t_{i}).$
(62)

Now applying $E\left(1+iJ(t_{i})\right)\hat{\varphi}$ to $\Psi_{vac}\in
H_{J(t_{i})}$ we get zero. Let $U(t_{f},t_{i})\Psi_{vac}=\psi\in
H_{J(t_{f})}$. We then get

$\displaystyle\left[\left(1+iEJ(t_{i})E^{-1}\right){\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}}\right]\psi=0.$
$\displaystyle\implies$
$\displaystyle\left[\left(1+iEJ(t_{i})E^{-1}\right)^{i}_{~{}\bar{j}}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{j}}}+\left(1+iEJ(t_{i})E^{-1})\right)^{i}_{~{}\underaccent{\bar}{j}}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{j}}}\right]\psi=0,$
$\displaystyle\implies$
$\displaystyle\left[\left(J(t_{f})+EJ(t_{i})E^{-1}\right)^{i}_{~{}\bar{j}}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{j}}}-\left(J(t_{f})-EJ(t_{i})E^{-1})\right)^{i}_{~{}\underaccent{\bar}{j}}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{j}}}\right]\psi=0,$
$\displaystyle\implies$
$\displaystyle\left[\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{i}}-\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\underaccent{\bar}{j}}\right]\psi=0,$
(63)

where $\chi$ is the operator
$\chi:=\left(J(t_{f})+EJ(t_{i})E^{-1}\right)^{-1}\left(J(t_{f})-EJ(t_{i})E^{-1}\right)$.
In the second equality above we have restored the indices and decomposed the
operator in terms of its holomorphic and anti-holomorphic components with
respect to $J(t_{f})$. In the third equality we have used the fact that
$J(t_{f})^{i}_{~{}\bar{j}}{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{j}}}=-i{\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\varphi}}}^{\bar{j}}}$
and similarly for the holomorphic component. But eq. 63 is the same as eq. 39
with the identification $J_{2}=J(t_{f})$ and $J_{1}=EJ(t_{i})E^{-1}$.
Therefore the result eq. 44 of sec. III.3 can be imported as is. We thus have
the conclusion that operator $U(t_{f},t_{i})$ is unitary iff

$\displaystyle\left[J(t_{f})-EJ(t_{i})E^{-1}\right]^{\bar{i}}_{~{}\underaccent{\bar}{k}}\left[J(t_{f})-EJ(t_{i})E^{-1}\right]^{\underaccent{\bar}{k}}_{~{}\bar{i}}<\infty.$
(64)

In our case, since our the complex structure $J(t_{f})$ is not specified
separately but is dictated by the time evolution of the initial complex
structure $J(t_{f})$, we have that $J(t_{f})=EJ(t_{i})E^{-1}$ and thus the
Hilbert-Schmidt condition is trivially satisfied. This matches with the
conclusion of ref. Agullo and Ashtekar (2015). Therefore, our evolution
operator $U$ is always unitary. Note that in the finite dimensional case there
is nothing to check because there will be finite number of terms in eq. 64 and
$U$ will be unitary. However, in the infinite dimensional case eq. 64 is a
non-trivial condition for ensuring the unitarity of Schrodinger dynamics.

What if one insists on specifying another complex structure on the final
slice, say, $J_{new}$? In that case our strategy would be to first use our
Schrodinger equation 50 to propagate the state till the final time slice to
get an element of $H_{J(t_{f})}$ and then do a Bogoliubov transformation to
get a state in $H_{J_{new}}$. From eq. 44 in sec. III.3 the latter will be
unitary iff $\left[J_{new}-J(t_{f})\right]$ is Hilbert-Schmidt, i.e., iff

$\displaystyle\left[J_{new}-J(t_{f})\right]^{\bar{i}}_{~{}\underaccent{\bar}{k}}\left[J_{new}-J(t_{f})\right]^{\underaccent{\bar}{k}}_{~{}\bar{i}}<\infty,$
(65)

where (anti-)holomorphic projections are with respect to $J_{new}$.

##  VI Summary and Outlook

Quantum field theory on curved spacetime is usually studied in the covariant
picture. In this picture the covariant operator algebra is given on the
spacetime. The choice of vacuum is equivalent to the choice of complex
structure on the covariant phase space. Once one chooses this complex
structure one can construct the wavefunctions which are holomorphic functions
with respect to this complex structure. Therefore the Hilbert space is labeled
by this complex structure and provides the representation space for the
representation of the covariant operator algebra. This is done either in terms
of creation/annihilation operators or by geometric quantization as discussed
in this paper. In practical applications one specifies the complex structure
by choosing a basis of the solutions of the linear field equation and
decomposing them in terms of positive and negative frequency. This induces a
holomorphic structure on the covariant phase space. In this approach dynamics
is carried by the operators and state is considered fixed. Correlation
functions of operators in this state are the observables of the theory. This
Heisenberg picture of quantum dynamics is sufficient and complete. However, if
one attempts to transfer the time dependence from the operators to the states
thus constructing a Schrodinger picture of quantum dynamics one runs into the
problem noted in refs. Torre and Varadarajan (1999); Helfer (1999a, b, 1996);
Cortez  _et al._ (2013a, b); Gomar  _et al._ (2012); Cortez  _et al._ (2012);
Gomez Vergel and Villasenor (2008); Barbero G.  _et al._ (2008); Corichi  _et
al._ (2007); Cortez  _et al._ (2007); Corichi  _et al._ (2006); Torre (2002)
that the Schrodinger dynamics turns out to be non-unitary. This issue was
resolved by Agullo and Ashtekar in ref. Agullo and Ashtekar (2015) who
realized that the problem arises because one is trying to construct the
Schrodinger dynamics on a fixed Hilbert space. If one insists on a fixed
Hilbert space then eq. 1 indeed leads to the conclusion that the operator $U$
is not unitary. However, if one allows the Hilbert space to change during time
evolution then ref. Agullo and Ashtekar (2015) postulated a generalized
unitarity condition stated in eq. 2 which leads to a unitary $U$. The goal of
our paper is to understand the operator $U$ that appears in eq. 2. In fact our
objective is complementary to that of eq. 2. We postulated the Schrodinger
dynamics directly by constructing the Hamiltonian operator in eq. 54 that
evolves the states infinitesimally. To this end we used the Bogoliubov
transformation implemented by the connection in eq. 32 first constructed in
refs. Witten (1993); Axelrod  _et al._ (1991). Operator $U$ is then the time-
ordered exponential of the Hamiltonian, eq. 52. We then derived the
generalized unitarity condition of ref. Agullo and Ashtekar (2015) and the
Hilbert-Schmidt condition that follows from it in sec. V. Given that we
postulated the Schrodinger equation, one might wonder why did we have to
derive the generalized unitarity condition before checking the unitarity of
the Schrodinger dynamics. Indeed, it would be nice to directly evolve the
vacuum wavefunction in eq. 19 using our $U$, but we found it simpler to first
derive the generalized unitarity condition.

Note that the condition for unitarity of our $U$ is that
$\left[J(t_{f})-EJ(t_{i})E^{-1}\right]$ be Hilbert-Schmidt. This is different
from the old unitarity condition which insisted on the dynamics on a fixed
Hilbert space $H_{J}$ and which leads to the condition that
$\left[J-EJE^{-1}\right]$ be Hilbert-Schmidt. Moreover, the condition is not
that $\left[J(t_{f})-J(t_{i})\right]$ be Hilbert-Schmidt, which would be the
case if were looking for a kinematical identification between the initial and
final Hilbert spaces as in sec. III.3. In fact, as shown in the cosmological
example in ref. Agullo and Ashtekar (2015), $\left[J(t_{f})-J(t_{i})\right]$
turns out not to be Hilbert-Schmidt. The latter in particular implies that in
the infinite dimensional setting, for arbitrary foliation of the background
spacetime, the connection $\mathcal{A}$ in eq. 32 is not unitary. It is only
that the dynamical evolution by our Hamiltonian, only a part of which is
$\mathcal{A}$, is unitary. This agrees with the insight of ref. Agullo and
Ashtekar (2015) where it was phrased by saying that only the dynamical
automorphisms of canonical operator algebra are unitarily implementable.

Finally we would like to make some remarks regarding the shortcomings of our
study. Clearly, it would be important to consider the infinite dimensional
language from the start, as it was in ref. Agullo and Ashtekar (2015).
Although phrasing the problem in the language of finite dimensions and taking
the limit to infinity is useful and physically insightful, there are issues
related to operator domains in Cauchy completions etc. which arise in infinite
dimensions that our analysis is completely oblivious of. We leave this problem
for those who are better equipped than us in the functional analytical issues
and mathematical rigor (see, for e.g., ref. Woodhouse (1981)). Another
important restriction is that, following refs. Witten (1993); Axelrod  _et
al._ (1991), our study was limited to translation invariant complex
structures. While this covers some known cases (see, for e.g., ref. Ashtekar
and Magnon (1975)), it certainly is not the most general situation. Last but
not least is the fact that the connection $\mathcal{A}$ is projectively flat.
Its curvature is $F=-\dfrac{1}{8}\delta
J^{\underaccent{\bar}{i}}_{~{}\bar{k}}\wedge\delta
J^{\bar{k}}_{~{}\underaccent{\bar}{i}}$. Finiteness of curvature looks like
the infinitesimal form of Hilbert-Schmidt condition. The connection can be
made flat by adding the so called metaplectic correction. In this paper, we
have not considered this. It would be interesting to understand the
metaplectic correction and its dynamical consequences in the context of
generalized unitarity.

###### Acknowledgements.

The work of MMK was supported by the grant provided by Region Bourgogne-
Franche-Comte under the program "Stages Monde". AM acknowledges useful
discussions with Rafi Alam and Sk. Noor Alam.

##  Appendix A Normalization of the state

State to be normalized:

$\displaystyle\psi=\mathscr{N}\,e^{-\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\varphi^{\bar{i}}\varphi^{\underaccent{\bar}{j}}}\,e^{\frac{i}{2}\omega_{\bar{i}\underaccent{\bar}{j}}\chi^{\bar{i}}_{~{}\underaccent{\bar}{k}}\varphi^{\underaccent{\bar}{k}}\varphi^{\underaccent{\bar}{j}}}.$
(66)

From eq. 20 we have the normalization condition,

$\displaystyle
1=|\mathscr{N}|^{2}\int\left(\prod_{i=1}^{2N}\,d\varphi^{i}\right)\sqrt{\det{\left(\frac{\omega}{2\pi}\right)}}\,\psi^{\ast}\psi.$
(67)

By explicitly inserting the holomorphic projections of eq. 15 in $\psi$ we get

$\displaystyle\psi=\mathscr{N}\,e^{-\frac{1}{4}\varphi^{i}\left(\omega
J_{2}\right)_{ij}\varphi^{j}}\,e^{-\frac{i}{4}\varphi^{i}\left(\omega\chi\right)_{ij}\varphi^{j}-\frac{1}{4}\varphi^{i}\left(\omega\chi
J_{2}\right)_{ij}\varphi^{j}}.$

Hence for $|\psi|^{2}$ we get

$\displaystyle|\psi|^{2}=|\mathscr{N}|^{2}e^{-\frac{1}{2}\varphi^{i}\left(\omega
J_{2}\right)_{ij}\varphi^{j}}\,e^{-\frac{1}{2}\varphi^{i}\left(\omega\chi
J_{2}\right)_{ij}\varphi^{j}}.$

The normalization equation has then just a Gaussian integral,

$\displaystyle
1=|\mathscr{N}|^{2}\int\left(\prod_{i=1}^{2N}\,d\varphi^{i}\right)\sqrt{\det{\left(\frac{\omega}{2\pi}\right)}}\,e^{-\frac{1}{2}\varphi^{i}M_{ij}\varphi^{j}},$

where $M$ is the matrix $M=\omega\left(1+\chi\right)J_{2}$. The $\det$ factor
in the above integral can come out of the integral since our $\omega$ is
translation invariant. The remaining Gaussian integral gives
$\frac{1}{\sqrt{\det\left(M/2\pi\right)}}$. A part of this cancels the
$\sqrt{\det\left(\omega/2\pi\right)}$ sitting outside and we are left with,

$\displaystyle
1=|\mathscr{N}|^{2}\frac{1}{\sqrt{\det\left[(1+\chi)J_{2}\right]}}$

Now, the matrix $\left[(1+\chi)J_{2}\right]$ can be decomposed in the
holomorphic and anti-holomorphic parts with respect to $J_{2}$, and noting
that $\chi$ has only mixed indices, we have

$\displaystyle\det\left[(1+\chi)J_{2}\right]$
$\displaystyle=\det\left[\begin{pmatrix}-i\delta^{\bar{i}}_{~{}\bar{j}}&0\\\
0&i\delta^{\underaccent{\bar}{i}}_{~{}\underaccent{\bar}{j}}\end{pmatrix}+\begin{pmatrix}0&i\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}\\\
-i\chi^{\underaccent{\bar}{i}}_{~{}\bar{j}}&0\end{pmatrix}\right]$ (68)
$\displaystyle=\det\left[\mathbbm{1}-\chi^{2}\right],$ (69)

where the matrix
$\chi^{2}:=\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}\chi^{\underaccent{\bar}{j}}_{~{}\bar{k}}$.
Note that $\chi^{\bar{i}}_{~{}\underaccent{\bar}{j}}$ and
$\chi^{\underaccent{\bar}{i}}_{~{}\bar{j}}$ are complex conjugate matrices
therefore $\chi^{2}$ is positive.

Now using that $\ln\det=\mathrm{tr}\ln$, we get

$\displaystyle\det\left[\mathbbm{1}-\chi^{2}\right]$
$\displaystyle=e^{\mathrm{tr}\ln\left[\mathbbm{1}-\chi^{2}\right]}$ (70)
$\displaystyle<e^{-\mathrm{tr}\left(\chi^{2}\right)}.$ (71)

Hence the normalization $\mathscr{N}$ is finite iff
$\mathrm{tr}\left(\chi^{2}\right)$ is finite. This is the case iff

$\displaystyle\left(J_{2}-J_{1}\right)^{\bar{i}}_{~{}\underaccent{\bar}{k}}\left(J_{2}-J_{1}\right)^{\underaccent{\bar}{k}}_{~{}\bar{i}}<\infty,$
(72)

which is the Hilbert-Schmidt condition.

## References

  * Wald (1995) Robert M. Wald,  _Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics_ , Chicago Lectures in Physics (University of Chicago Press, Chicago, IL, 1995).
  * Ashtekar and Magnon (1975) A. Ashtekar and A. Magnon, “Quantum Fields in Curved Space-Times,” Proc. Roy. Soc. Lond. A 346, 375–394 (1975).
  * Ashtekar and Magnon-Ashtekar (1980) A. Ashtekar and A. Magnon-Ashtekar, “A Geometrical Approach to External Potential Problems in Quantum Field Theory,” Gen. Rel. Grav. 12, 205–223 (1980).
  * Torre and Varadarajan (1999) Charles G. Torre and Madhavan Varadarajan, “Functional evolution of free quantum fields,” Class. Quant. Grav. 16, 2651–2668 (1999), arXiv:hep-th/9811222 .
  * Helfer (1999a) Adam D. Helfer, “The Hamiltonians of linear quantum fields. 1. Existence theory,” (1999a), arXiv:hep-th/9908011 .
  * Helfer (1999b) Adam D. Helfer, “The Hamiltonians of linear quantum fields. 2. Classically positive Hamiltonians,” (1999b), arXiv:hep-th/9908012 .
  * Helfer (1996) Adam D. Helfer, “The Stress - energy operator,” Class. Quant. Grav. 13, L129–L134 (1996), arXiv:gr-qc/9602060 .
  * Cortez  _et al._ (2013a) Jeronimo Cortez, Daniel Martin-de Blas, Guillermo A. Mena Marugan, and Jose Velhinho, “Massless scalar field in de Sitter spacetime: unitary quantum time evolution,” Class. Quant. Grav. 30, 075015 (2013a), arXiv:1301.4920 [gr-qc] .
  * Cortez  _et al._ (2013b) Jerónimo Cortez, Lucía Fonseca, Daniel Martín-de Blas, and Guillermo A. Mena Marugán, “Uniqueness of the Fock quantization of scalar fields under mode preserving canonical transformations varying in time,” Phys. Rev. D 87, 044013 (2013b), arXiv:1212.3947 [gr-qc] .
  * Gomar  _et al._ (2012) Laura Castello Gomar, Jeronimo Cortez, Daniel Martin-de Blas, Guillermo A. Mena Marugan, and Jose M. Velhinho, “Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes,” JCAP 11, 001 (2012), arXiv:1211.5176 [gr-qc] .
  * Cortez  _et al._ (2012) Jeronimo Cortez, Guillermo A. Mena Marugan, Javier Olmedo, and Jose M. Velhinho, “Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields,” Phys. Rev. D 86, 104003 (2012), arXiv:1202.6330 [gr-qc] .
  * Gomez Vergel and Villasenor (2008) Daniel Gomez Vergel and Eduardo J. S. Villasenor, “Unitary evolution of free massless fields in de Sitter space-time,” Class. Quant. Grav. 25, 145008 (2008), arXiv:0712.1421 [gr-qc] .
  * Barbero G.  _et al._ (2008) J. Fernando Barbero G., Daniel Gomez Vergel, and Eduardo J. S. Villasenor, “Quantum unitary evolution of linearly polarized S**1 x S**2 and S**3 Gowdy models coupled to massless scalar fields,” Class. Quant. Grav. 25, 085002 (2008), arXiv:0711.1790 [gr-qc] .
  * Corichi  _et al._ (2007) Alejandro Corichi, Jeronimo Cortez, Guillermo A. Mena Marugan, and Jose M. Velhinho, “Quantum Gowdy T**3 model: Schrodinger representation with unitary dynamics,” Phys. Rev. D 76, 124031 (2007), arXiv:0710.0277 [gr-qc] .
  * Cortez  _et al._ (2007) Jeronimo Cortez, Guillermo A. Mena Marugan, and Jose M. Velhinho, “Uniqueness of the Fock quantization of the Gowdy T**3 model,” Phys. Rev. D 75, 084027 (2007), arXiv:gr-qc/0702117 .
  * Corichi  _et al._ (2006) Alejandro Corichi, Jeronimo Cortez, Guillermo A. Mena Marugan, and Jose M. Velhinho, “Quantum Gowdy T**3 model: A Uniqueness result,” Class. Quant. Grav. 23, 6301–6320 (2006), arXiv:gr-qc/0607136 .
  * Torre (2002) C. G. Torre, “Quantum dynamics of the polarized Gowdy T**3 model,” Phys. Rev. D 66, 084017 (2002), arXiv:gr-qc/0206083 .
  * Agullo and Ashtekar (2015) Ivan Agullo and Abhay Ashtekar, “Unitarity and ultraviolet regularity in cosmology,” Phys. Rev. D 91, 124010 (2015), arXiv:1503.03407 [gr-qc] .
  * Parker (1968) L. Parker, “Particle creation in expanding universes,” Phys. Rev. Lett. 21, 562–564 (1968).
  * Parker (1969) Leonard Parker, “Quantized fields and particle creation in expanding universes. 1.” Phys. Rev. 183, 1057–1068 (1969).
  * Parker (1971) L. Parker, “Quantized fields and particle creation in expanding universes. 2.” Phys. Rev. D 3, 346–356 (1971), [Erratum: Phys.Rev.D 3, 2546–2546 (1971)].
  * Witten (1993) Edward Witten, “Quantum background independence in string theory,” in  _Conference on Highlights of Particle and Condensed Matter Physics (SALAMFEST)_ (1993) arXiv:hep-th/9306122 .
  * Axelrod  _et al._ (1991) Scott Axelrod, Steve Della Pietra, and Edward Witten, “Geometric quantization of Chern-Simons gauge theory,” J. Diff. Geom. 33, 787–902 (1991).
  * Nair (2016) V. P. Nair, “Elements of Geometric Quantization and Applications to Fields and Fluids,” (2016), arXiv:1606.06407 [hep-th] .
  * Woodhouse (1992) N. M. J. Woodhouse,  _Geometric quantization_ (1992).
  * Woodhouse (1981) N. Woodhouse, “Geometric Quantization and the Bogoliubov Transformation,” Proceedings of the Royal Society of London Series A 378, 119–139 (1981).