$\|\cdot\|_{\beta,\bullet}$ and $\|\cdot\|_{-\beta,\bullet}$ on
${\mathcal{W}}_{\beta}$, resp. ${\mathcal{W}}_{-\beta}$ there exists
$C_{\bullet}$ such that

$\lvert(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)\rvert\leq C_{\bullet}\lVert
v\rVert_{\beta,\bullet}\lVert w\rVert_{-\beta,\bullet}.$ (4.54)

In what follows, let
$({\mathcal{W}}_{-\frac{1}{2}},{\mathcal{W}}_{\frac{1}{2}},Q)$ be a nested
pre-pseudo-unitary pair. (The parameter $\frac{1}{2}$ can be changed to any
positive number, it is chosen here in view of our future applications). Let
${\mathcal{W}}_{\beta}$, $\beta\in[-\frac{1}{2},\frac{1}{2}]$, be the
Hilbertizable spaces obtained by interpolation from the nested pair
$({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}})$.

######  Proposition 4.39.

For $\beta\in[-\frac{1}{2},\frac{1}{2}]$, there exists a unique family of
Hermitian pairings $Q_{\beta}$

${\mathcal{W}}_{\beta}\times{\mathcal{W}}_{-\beta}\ni(v,w)\mapsto(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}Q_{\beta}w)\in{{\mathbb{C}}}.$ (4.55)

such that $Q_{\frac{1}{2}}=Q$ and if
$-\frac{1}{2}\leq\beta_{1}\leq\beta_{2}\leq\frac{1}{2}$,
$v\in{\mathcal{W}}_{-\beta_{1}}\subset{\mathcal{W}}_{-\beta_{2}}$,
$w\in{\mathcal{W}}_{\beta_{2}}\subset{\mathcal{W}}_{\beta_{1}}$, then

$(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Q_{\beta_{1}}w)=(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q_{\beta_{2}}w).$ (4.56)

###### Proof.

By (4.54), $Q$ can be viewed as a bounded map from
${\mathcal{W}}_{-\frac{1}{2}}$ to the antidual of
${\mathcal{W}}_{\frac{1}{2}}$. The antidual of ${\mathcal{W}}_{\frac{1}{2}}$
coincides with ${\mathcal{W}}_{-\frac{1}{2}}$. Hence, $Q\in
B({\mathcal{W}}_{-\frac{1}{2}})$.

The restriction of $Q$ to ${\mathcal{W}}_{\frac{1}{2}}$, by (4.54) and (4.53)
is a bounded map to the antidual of ${\mathcal{W}}_{-\frac{1}{2}}$, which is
${\mathcal{W}}_{\frac{1}{2}}$. Hence $Q\in B({\mathcal{W}}_{\frac{1}{2}})$.

By interpolation, that is, Prop. 2.1 (2), for
$\beta\in[-\frac{1}{2},\frac{1}{2}]$, the restriction of $Q$ to
${\mathcal{W}}_{\beta}$ is bounded. ∎

In what follows we drop the subscript $\beta$ from $Q_{\beta}$, which is
allowed because of (4.56). In particular, for $\beta=0$, we obtain a bounded
Hermitian form on ${\mathcal{W}}_{0}$:

${\mathcal{W}}_{0}\times{\mathcal{W}}_{0}\ni(v,w)\mapsto(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw)\in{{\mathbb{C}}}.$

###  4.11 Evolutions on nested pre-pseudo-unitary pairs

In this subsection we suppose that
$({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}},Q)$ is a nested
pre-pseudo-unitary pair.

######  Definition 4.40.

If a bounded operator $R$ on ${\mathcal{W}}_{-\frac{1}{2}}$, restricts to a
bounded operator on ${\mathcal{W}}_{\frac{1}{2}}$, and

$(Rv\mkern 2.0mu|\mkern 2.0mu\mathopen{}QRw)=(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw),\quad v\in{\mathcal{W}}_{-\frac{1}{2}},\quad
w\in{\mathcal{W}}_{\frac{1}{2}},$

then we say that _$R$ preserves
$({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}},Q)$_.

Applying complex interpolation we obtain

######  Proposition 4.41.

Suppose that $R$ preserves
$({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}},Q)$. Then for
$0\leq\beta\leq\frac{1}{2}$ it restricts to an operator preserving
$({\mathcal{W}}_{\beta},{\mathcal{W}}_{-\beta},Q)$. In particular, $R$
preserves $Q$ on ${\mathcal{W}}_{0}$ in the usual sense.

######  Definition 4.42.

Suppose that $B$ is an operator on ${\mathcal{W}}_{-\frac{1}{2}}$ with domain
containing ${\mathcal{W}}_{\frac{1}{2}}$. We say that $B$ _infinitesimally
preserves_ $({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}},Q)$ if
$B$ is a generator of a group on ${\mathcal{W}}_{-\frac{1}{2}}$, its part
$\tilde{B}$ in ${\mathcal{W}}_{\frac{1}{2}}$ is a generator of a group on
${\mathcal{W}}_{\frac{1}{2}}$, and

$(Bv\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)=(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}QBw),\quad v,w\in{\mathcal{W}}_{\frac{1}{2}}.$ (4.57)

The quadratic form defined by (4.57) is called the _energy_ or _Hamiltonian
quadratic form of $B$ on ${\mathcal{W}}_{\frac{1}{2}}$_.

######  Proposition 4.43.

Suppose that $B$ is an operator on ${\mathcal{W}}_{-\frac{1}{2}}$ that
infinitesimally preserves
$({\mathcal{W}}_{-\frac{1}{2}},{\mathcal{W}}_{\frac{1}{2}},Q)$. Then
$\mathrm{e}^{-\mathrm{i}tB}$, $t\in{{\mathbb{R}}}$, preserves
$({\mathcal{W}}_{-\frac{1}{2}},{\mathcal{W}}_{\frac{1}{2}},Q)$.

The following theorem can be viewed as an alternative to Thm. 4.31:

######  Theorem 4.44.

Suppose that all assumptions of Thm. 3.12 are satisfied where ${\mathcal{W}}$
is replaced with ${\mathcal{W}}_{-\frac{1}{2}}$ and ${\mathcal{V}}$ is
replaced with ${\mathcal{W}}_{\frac{1}{2}}$. In addition, assume that for all
$t\in I$ the operators $B(t)$ infinitesimally preserve
$({\mathcal{W}}_{-\frac{1}{2}},{\mathcal{W}}_{\frac{1}{2}},Q)$. Then the
evolution $R(t,s)$ preserves
$({\mathcal{W}}_{-\frac{1}{2}},{\mathcal{W}}_{\frac{1}{2}},Q)$. In particular,
it is pre-pseudo-unitary on ${\mathcal{W}}_{0}$.

Let us compare the constructions of Thm. 4.31 and of Theorem 4.44. In both
cases we obtain a (pre-)pseudo-unitary evolution on a (pre-)pseudo-unitary
space. However, in the former case we have a fixed space ${\mathcal{V}}$
contained in $\operatorname{{\mathcal{D}}}(B(t))$ for all $t$. In the latter
case, we do not have information about the domain of the generator of the
evolution on ${\mathcal{W}}_{0}$, that is, of the part of $B(t)$ in
${\mathcal{W}}_{0}$. On the other hand, in practice the assumptions of Theorem
4.44 can be weaker.

##  5 Abstract Klein-Gordon operator

The usual Klein-Gordon operator acts on, say, $C_{\mathrm{c}}^{\infty}(M)$,
where $M$ is a Lorentzian manifold, and is given by the expression (1.5). One
of the main ideas of our paper says one should interpreted $K$ as a Hermitian
operator in the sense of the Hilbert space $L^{2}(M)$. After identification of
$M$ with $I\times\Sigma$, where $I$ corresponds to the time variable and
$\Sigma$ is describes the spatial variables, we can identify $L^{2}(M)$ with
$L^{2}(I,{\mathcal{K}})$, where ${\mathcal{K}}=L^{2}(\Sigma)$. Then formally,
the operator $K$ is given by the expression

$K:=\big{(}D_{t}+W^{*}(t)\big{)}\frac{1}{\alpha^{2}(t)}\big{(}D_{t}+W(t)\big{)}-L(t),$
(5.1)

where $W(t)$ consists mostly of the $0$th component of the potential and
$L(t)$ is the magnetic Schrödinger operator on $\Sigma$. We will describe this
identification in more detail in Subsect. 8.3.

In this section we study (5.1) in an abstract setting. We are interested in
various inverses and bisolutions of $K$. We treat ${\mathcal{K}}$ as an
abstract Hilbert space and $L(t)$, $\alpha(t)$, $W(t)$ as given abstract
operators. The results of this section will be applied to the true Klein-
Gordon operator in Sect. 8.

In order to study propagators associated with the abstract Klein-Gordon
operator, we first introduce a certain scale of Hilbertizable spaces
${\mathcal{W}}_{\lambda}$ represented as the direct sum of the “configuration
space” and the “momentum space”, made out of appropriate Sobolev-type spaces
based on ${\mathcal{K}}$. The space ${\mathcal{W}}_{0}$ has the structure of a
Krein space and will play the central role in quantization. We construct a
certain pseudo-unitary dynamics on ${\mathcal{W}}_{0}$ associated with $K$
with the generator $B(t)$ made out of $L(t),W(t),\alpha^{2}(t)$. After
imposing boundary conditions we can define propagators, first associated with
the evolution equation operator $M=\partial_{t}+\mathrm{i}B(t)$, and then with
the operator $K$ itself.

Note that the formula (5.1) does not give a rigorous definition of a unique
closed operator. Actually, the analysis of possible closed realizations of $K$
and the corresponding inverses is quite subtle and depends strongly on whether
$I$ is finite or not.

In the case of a finite $I$ one needs first to impose appropriate boundary
conditions at the initial time $t_{-}$ and the final time $t_{+}$. This leads
to a construction of an inverses of the evolution equation operator, from
which an inverse of the Klein-Gordon operator is obtained. Both inverses are
bounded. Inverting them we obtain well-posed a realization of the Klein-Gordon
operator.

The situation is different and less understood if $I=\mathbb{R}$. The Feynman
and anti-Feynman inverse can be constructed, however they are not bounded on
$L^{2}(\mathrm{R},{\mathcal{K}})$. We conjecture that there exists a
distinguished self-adjoint realization $K^{\mathrm{s.a.}}$ of $K$ such that
these inverses are the boundary values of the resolvent of $K^{\mathrm{s.a.}}$
from above and below at zero. We know that this conjecture is true in some
special cases, notably if $K$ is stationary. We describe some arguments in
favor of the conjecture, notably, we sketch a possible construction of the
resolvent.

Throughout the section we need to overcome a number of technical issues.
First, it is convenient to assume that the Hamiltonian used in the
construction of the phase space is bounded away from zero, or in physical
terms, that the mass is strictly positive. However, physical systems may have
a zero mass. This is solved by assuming that the phase space is constructed
not directly from the Hamiltonian $H(t)$, but from $H_{0}(t)$ which differs by
a constant $b$.

Another problem is the regularity. Assumption 5.1 allows us to perform our
basic construction, however does not suffice to prove some desired properties
of the propaagators. This problem essentially disappears when we assume that
everything is smooth—therefore it can be considered as purely academic, of
interest only to specialists in operator theory.

###  5.1 Abstract Klein-Gordon form

Throughout the section we assume that ${\mathcal{K}}$ is a Hilbert space and
for $t\in I^{\mathrm{cl}}$ we are given the following operators on
${\mathcal{K}}$:

  1. (1)

self-adjoint $L(t)$ for which there exists $b\in\mathbb{R}$ such that
$L_{0}(t):=L(t)+b$ are positive invertible,

  2. (2)

bounded invertible selfadjoint $\alpha(t)$,

  3. (3)

$W(t)$.

We will say that an operator function $I\ni t\mapsto A(t)\in B({\mathcal{K}})$
is absolutely norm continuous if there exists $c\in L^{1}(I)$ such that $c\geq
0$ and

$\|A(t)-A(s)\|\leq\int_{t}^{s}c(\tau)\mathop{}\\!\mathrm{d}\tau,\quad t\leq
s,\quad t,s\in I.$ (5.2)

Here is the basic assumption that we will use in this section.

######  Assumption 5.1.

  1. (1)

For any $t\in I$ there exist $c_{1},c_{2}$ such that

$c_{1}L_{0}(0)\leq L_{0}(t)\leq c_{2}L_{0}(0)$ (5.3)

and $I^{\mathrm{cl}}\ni t\mapsto
L_{0}(0)^{-\frac{1}{2}}L_{0}(t)L_{0}(0)^{-\frac{1}{2}}\in B({\mathcal{K}})$ is
absolutely norm continuous.

  2. (2)

$I^{\mathrm{cl}}\ni t\mapsto\alpha^{2}(t)\in B({\mathcal{K}})$ is an
absolutely norm continuous.

  3. (3)

$I^{\mathrm{cl}}\ni t\mapsto W(t)L_{0}(t)^{-\frac{1}{2}}\in B({\mathcal{K}})$
is absolutely norm continuous and there exists $a<1$ such that

$|(f|W(t)g)|\leq a\|\alpha(t)f\|\|L_{0}(t)^{\frac{1}{2}}g\|.$ (5.4)

Define the scales of Hilbert spaces
${\mathcal{K}}_{\beta,t}:=L_{0}(t)^{-\frac{\beta}{2}}{\mathcal{K}}$. Then by
(5.3) for $\beta\in[-1,1]$ ${\mathcal{K}}_{\beta,t}$ is compatible with
${\mathcal{K}}_{\beta,0}$. The corresponding scale of Hilbertizable spaces
will be denoted ${\mathcal{K}}_{\beta}$, $\beta\in[-1,1]$.

For $f_{1},f_{2}\in C_{\mathrm{c}}(I,{\mathcal{K}}_{1})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{K}}_{0})$ we define the Klein-Gordon quadratic
form

$\displaystyle(f_{1}|Kf_{2})$
$\displaystyle=\int_{I}\Big{(}\big{(}D_{t}+W(t)\big{)}f_{1}(t)|\frac{1}{\alpha^{2}(t)}\big{(}D_{t}+W(t)\big{)}f_{2}(t)\Big{)}\mathop{}\\!\mathrm{d}t$
$\displaystyle-\int_{I}\big{(}L_{0}(t)^{\frac{1}{2}}f_{1}(t)|L_{0}(t)^{\frac{1}{2}}f_{2}(t)\big{)}\mathop{}\\!\mathrm{d}t+b\int_{I}\big{(}f_{1}(t)|f_{2}(t)\big{)}\mathop{}\\!\mathrm{d}t.$
(5.5)

Note that $K$ is a Hermitian form in the sense of the Hilbert space
$L^{2}(I)\otimes{\mathcal{K}}\simeq L^{2}(I,{\mathcal{K}})$. Formally, it
corresponds to the operator given by the expression 5.1.

We would like to interpret $K$ as a closed operator on
$L^{2}(I,{\mathcal{K}})$. Unfortunately, $K$ is not semibounded, hence the
usual theory of quadratic forms does not apply.

###  5.2 Pseudo-unitary evolution on the space of Cauchy data

The following analysis is essentially an adaptation of [6, 7] to the abstract
setting.

We consider the scale of Hilbert spaces

${\mathcal{W}}_{\lambda}={\mathcal{K}}_{\lambda+\frac{1}{2}}\oplus{\mathcal{K}}_{\lambda-\frac{1}{2}},\quad\lambda\in\left[-\frac{1}{2},\frac{1}{2}\right].$

Of special importance are the spaces

$\displaystyle{\mathcal{W}}_{\mathrm{en}}$
$\displaystyle:={\mathcal{K}}_{1}\oplus{\mathcal{K}}_{0}$
$\displaystyle={\mathcal{W}}_{\frac{1}{2}},$ (5.6)
$\displaystyle{\mathcal{W}}_{\mathrm{dyn}}$
$\displaystyle:={\mathcal{K}}_{\frac{1}{2}}\oplus{\mathcal{K}}_{-\frac{1}{2}}$
$\displaystyle={\mathcal{W}}_{0},$ (5.7)
$\displaystyle{\mathcal{W}}_{\mathrm{en}}^{*}$
$\displaystyle:={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{-1}$
$\displaystyle={\mathcal{W}}_{-\frac{1}{2}}.$ (5.8)

For $\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}\in{\mathcal{W}}_{-\frac{1}{2}}$ and
$\begin{bmatrix}v_{1}\\\ v_{2}\end{bmatrix}\in{\mathcal{W}}_{\frac{1}{2}}$ we
introduce the charge pairing defined by the charge operator
$Q=\begin{bmatrix}0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\\\ {{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\end{bmatrix}$. In
other words,

$\Big{(}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}\Big{|}Q\begin{bmatrix}v_{1}\\\
v_{2}\end{bmatrix}\Big{)}=(u_{1}|v_{2})+(u_{2}|v_{1}).$

Thus $({\mathcal{W}}_{-\frac{1}{2}},{\mathcal{W}}_{\frac{1}{2}},Q)$ is a Krein
nested triple and $({\mathcal{W}}_{0},Q)$ is a Krein space.

Set

$\displaystyle H(t)$ $\displaystyle=\begin{bmatrix}L(t)&W^{*}(t)\\\
W(t)&\alpha^{2}(t)\end{bmatrix},\quad
H_{0}(t)=\begin{bmatrix}L_{0}(t)&W^{*}(t)\\\ W(t)&\alpha^{2}(t)\end{bmatrix}.$
(5.9)

######  Proposition 5.2.

Suppose Assumption 5.1 holds. Then $H_{0}(t)$ is a positive operator on
${\mathcal{K}}\oplus{\mathcal{K}}$ with the form domain
${\mathcal{W}}_{\frac{1}{2}}$. Moreover, there exist $0<c_{1}\leq c_{2}$ such
that

$\displaystyle c_{1}\begin{bmatrix}L_{0}(t)&0\\\ 0&{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\leq$ $\displaystyle H_{0}(t)\leq
c_{2}\begin{bmatrix}L_{0}(t)&0\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix},$
(5.10) $\displaystyle c_{1}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ 0&L_{0}(t)\end{bmatrix}\leq$ $\displaystyle
QH_{0}(t)Q\leq c_{2}\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
0&L_{0}(t)\end{bmatrix}.$ (5.11)

Therefore,

$\displaystyle{\mathcal{W}}_{\frac{1}{2},t}:=H_{0}(t)^{-\frac{1}{2}}({\mathcal{K}}\oplus{\mathcal{K}})$
$\displaystyle\quad\text{ is compatible with }{\mathcal{W}}_{\frac{1}{2}},$
(5.12)
$\displaystyle{\mathcal{W}}_{-\frac{1}{2},t}:=(QH_{0}(t)Q)^{\frac{1}{2}}({\mathcal{K}}\oplus{\mathcal{K}})$
$\displaystyle\quad\text{ is compatible with }{\mathcal{W}}_{-\frac{1}{2}}.$
(5.13)

Let ${\mathcal{W}}_{\lambda,t}$, $\lambda\in\mathbb{R}$, denote the scale of
Hilbert spaces defined by interpolation from the nested pair
${\mathcal{W}}_{\frac{1}{2},t}$, ${\mathcal{W}}_{-\frac{1}{2},t}$. Clearly,
${\mathcal{W}}_{\lambda,t}$ are compatible with the Hilbertizable spaces
${\mathcal{W}}_{\lambda}$ for $\lambda\in[-1,1]$.

Set

$\displaystyle B(t)$
$\displaystyle:=QH(t)=\begin{bmatrix}W(t)&\alpha^{2}(t)\\\
L(t)&W^{*}(t)\end{bmatrix},\quad
B_{0}(t):=QH_{0}(t)=\begin{bmatrix}W(t)&\alpha^{2}(t)\\\
L_{0}(t)&W^{*}(t)\end{bmatrix}.$ (5.14)

######  Proposition 5.3.

Suppose Assumption 5.1 holds. Then for any $\lambda\in\mathbb{R}$, $B_{0}(t)$
is a unitary operator from ${\mathcal{W}}_{\frac{1}{2}+\lambda,t}$ to
${\mathcal{W}}_{-\frac{1}{2}+\lambda,t}$. Besides, it is a self-adjoint
operator in the sense of ${\mathcal{W}}_{-\frac{1}{2}+\lambda,t}$ with the
domain ${\mathcal{W}}_{\frac{1}{2}+\lambda,t}$. Therefore,

${\mathcal{W}}_{\lambda,t}=|B_{0}(t)|^{-\lambda+\frac{1}{2}}{\mathcal{W}}_{\frac{1}{2},t}=|B_{0}(t)|^{-\lambda-\frac{1}{2}}{\mathcal{W}}_{-\frac{1}{2},t}.$
(5.15)

###### Proof.

We drop $0$ and $t$ from $H_{0}(t),B_{0}(t)$. First note that $H$ is bounded
from ${\mathcal{K}}_{\frac{1}{2}}\oplus{\mathcal{K}}_{0}$ to
${\mathcal{K}}_{-\frac{1}{2}}\oplus{\mathcal{K}}_{0}$. Hence $B$ is bounded
from
${\mathcal{W}}_{\frac{1}{2}}={\mathcal{K}}_{\frac{1}{2}}\oplus{\mathcal{K}}_{0}$
to
${\mathcal{W}}_{-\frac{1}{2}}={\mathcal{K}}_{0}\oplus{\mathcal{K}}_{-\frac{1}{2}}$.
Now,

$(Bu|Bv)_{-\frac{1}{2},t}=(QHu|(QHQ)^{-1}QHv)=(u|Hv)=(u|v)_{\frac{1}{2},t}.$
(5.16)

This proves the unitarity from ${\mathcal{W}}_{\frac{1}{2},t}$ to
${\mathcal{W}}_{-\frac{1}{2},t}$.

Let $u,v\in{\mathcal{W}}_{\frac{1}{2}}$.

$\displaystyle(u|Bv)_{-\frac{1}{2},t}=(u|(QHQ)^{-1}QHv)=(QHu|(QHQ)^{-1}v)=(Bu|v)_{-\frac{1}{2},t}$
(5.17)

proves the Hermiticity in the sense of ${\mathcal{W}}_{-\frac{1}{2},t}$ with
the domain ${\mathcal{W}}_{\frac{1}{2},t}$. Clearly, a unitary Hermitian
operator is self-adjoint.

Now we obtain

${\mathcal{W}}_{-\frac{1}{2},t}=|B|{\mathcal{W}}_{\frac{1}{2},t},$ (5.18)

from which (5.15) and all the remaining statements of the proposition follow.
∎

######  Proposition 5.4.

Suppose Assumption 5.1 hold. Then $I\ni t\mapsto
B(t):{\mathcal{W}}_{\frac{1}{2}}\to{\mathcal{W}}_{-\frac{1}{2}}$ satisfies the
assumptions of Thm 4.44. Therefore, it defines an evolution $R(t,s)$ on
${\mathcal{W}}_{\lambda}$, $-\frac{1}{2}\leq\lambda\leq\frac{1}{2}$, pseudo-
unitary in the sense of
$({\mathcal{W}}_{-\lambda},{\mathcal{W}}_{\lambda},Q)$. In particular, the
evolution $R(t,s)$ is pseudo-unitary on ${\mathcal{W}}_{0}$ in the usual
sense.

###### Proof.

First we check that $B_{0}(t)$ satisfies the assumptions of Theorem 3.11 with
the space ${\mathcal{W}}={\mathcal{W}}_{-\frac{1}{2}}$ and
${\mathcal{V}}={\mathcal{W}}_{\frac{1}{2}}$, and the scalar products
$(\cdot|\cdot)_{-\frac{1}{2},t}$ and $(\cdot|\cdot)_{\frac{1}{2},t}$. Note
that for all $t\in I^{\mathrm{cl}}$ ${\mathcal{W}}_{\frac{1}{2}}$ is the
domain of $B_{0}(t)$ viewed as an operator on ${\mathcal{W}}_{-\frac{1}{2}}$.
Besides, $t\mapsto B_{0}(t)$ is norm continuous in
$B({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}})$.

By Prop. 5.3 $B(t)$ is self-adjoint in the sense of both
${\mathcal{W}}_{\frac{1}{2},t}$ and ${\mathcal{W}}_{-\frac{1}{2},t}$.

There exists $c\in L^{1}$ such that for $t,s\in I^{\mathrm{cl}}$,

$\left\|(H_{0}(t))^{\frac{1}{2}}(H_{0}(s))^{-\frac{1}{2}}v\right\|=\left\|(QH_{0}(t)Q)^{\frac{1}{2}}(QH_{0}(s)Q)^{-\frac{1}{2}}v\right\|\leq\exp\left|\int_{t}^{s}c(\tau)\mathop{}\\!\mathrm{d}\tau\right|.$
(5.19)

$\displaystyle\|w\|_{\frac{1}{2},s}$
$\displaystyle=\left\|H_{0}(t)^{-\frac{1}{2}}H_{0}(s)^{\frac{1}{2}}v\right\|_{\frac{1}{2},t}\leq\exp\left(\int_{t}^{s}c(\tau)\mathop{}\\!\mathrm{d}\tau\right)\|v\|_{\frac{1}{2},t},$
(5.20) $\displaystyle\|w\|_{-\frac{1}{2},s}$
$\displaystyle=\left\|(QH_{0}(t)Q)^{\frac{1}{2}}(QH_{0}(s)Q)^{-\frac{1}{2}}v\right\|_{-\frac{1}{2},t}\leq\exp\left(\int_{t}^{s}c(\tau)\mathop{}\\!\mathrm{d}\tau\right)\|v\|_{-\frac{1}{2},t}.$
(5.21)

The perturbation $B(t)-B_{0}(t)$ is bounded. Therefore, the assumptions of Thm
3.12 are satisfied.

Finally, both $B_{0}(t)$ and $B(t)$ infinitesimally preserve the Krein nested
triple $({\mathcal{W}}_{\frac{1}{2}},{\mathcal{W}}_{-\frac{1}{2}},Q)$. Hence
the assumptions of Thm 4.44 hold. ∎

###  5.3 Additional regularity of the Klein-Gordon operator

It will be useful to introduce a family of assumptions more restrictive than
Assumption 5.1. It will depend on a parameter $\rho\geq 0$.

######  Assumption 5.5 ($\rho$).

  1. (1)

For any $t\in I$ there exist $c_{1},c_{2}$ such that

$c_{1}L_{0}(0)^{1+\rho}\leq L_{0}(t)^{1+\rho}\leq c_{2}L_{0}(0)^{1+\rho}$
(5.22)

and $I^{\mathrm{cl}}\ni t\mapsto
L_{0}(0)^{-\frac{1+\rho}{2}}L_{0}(t)^{1+\rho}L_{0}(0)^{-\frac{1+\rho}{2}}\in
B({\mathcal{K}})$ is absolutely norm continuous.

  2. (2)

$I^{\mathrm{cl}}\ni t\mapsto
L_{0}(t)^{\frac{\rho}{2}}\alpha^{2}(t)L_{0}(t)^{-\frac{\rho}{2}}\in
B({\mathcal{K}})$ is absolutely norm continuous.

  3. (3)

$I^{\mathrm{cl}}\ni t\mapsto
L_{0}(t)^{\frac{\rho}{2}}W(t)L_{0}(t)^{\frac{-1-\rho}{2}},\quad
L_{0}(t)^{-\frac{\rho}{2}}W(t)L_{0}(t)^{\frac{-1+\rho}{2}}\in
B({\mathcal{K}})$ are absolutely norm continuous and there exists $a<1$ such
that

$|(f|W(t)g)|\leq
a\|\alpha(t)L(t)^{\frac{\rho}{2}}f\|\|L_{0}(t)^{\frac{1-\rho}{2}}g\|.$ (5.23)

Note that Assumption 5.1 coincides with Assumption 5.5($0$). If
$0\leq\rho^{\prime}\leq\rho$, then Assumption 5.5($\rho$) implies Assumption
5.5($\rho^{\prime}$).

######  Proposition 5.6.

Suppose Assumption 5.5($\rho$) holds. Then the following is true:

  1. (1)

For any $t\in I^{\mathrm{cl}}$ and for $|\beta|\leq 1+\rho$, the Hilbert
spaces ${\mathcal{K}}_{\beta,t}$ and ${\mathcal{K}}_{\beta,0}$ are compatible.
Hence we can extend the scale of Hilbertizable spaces ${\mathcal{K}}_{\beta}$
to $|\beta|\leq 1+\rho$, and also the scale of spaces
${\mathcal{W}}_{\lambda}$ to $|\lambda|\leq\frac{1}{2}+\rho$.

  2. (2)

For any $t\in I^{\mathrm{cl}}$ and $|\lambda|\leq\frac{1}{2}+\rho$, the
Hilbert spaces ${\mathcal{W}}_{\lambda,t}$ are compatible with
${\mathcal{W}}_{\lambda}$.

  3. (3)

For any $|\theta|\leq\rho$ the function $I\ni t\mapsto
B(t):{\mathcal{W}}_{\theta+\frac{1}{2}}\to{\mathcal{W}}_{\theta-\frac{1}{2}}$
satisfies the assumptions of Theorem 3.12, and hence defines an evolution
$R(t,s)$ on ${\mathcal{W}}_{\lambda}$, $|\lambda|\leq\frac{1}{2}+\rho$
satisfying

$\big{(}\partial_{t}+\mathrm{i}B(t)\big{)}R(t,s)u=0,\quad
u\in{\mathcal{W}}_{\theta+\frac{1}{2}}.$ (5.24)

###### Proof.

Assumption 5.5($\rho$) implies immediately that the Hilbertizable spaces
${\mathcal{K}}_{\pm(1+\rho),t}$ and ${\mathcal{K}}_{\pm(1+\rho),0}$ coincide.
Therefore (1) follows by by the Kato-Heinz inequality.

Let us drop $(t)$. We have

$\displaystyle\begin{bmatrix}L_{0}^{\frac{\theta}{2}}&0\\\
0&L_{0}^{\frac{-1+\theta}{2}}\end{bmatrix}\begin{bmatrix}W&\alpha^{2}\\\
L_{0}&W^{*}\end{bmatrix}\begin{bmatrix}L_{0}^{\frac{-1-\theta}{2}}&0\\\
0&L_{0}^{-\frac{\theta}{2}}\end{bmatrix}$ (5.25)
$\displaystyle=\begin{bmatrix}L_{0}^{\frac{\theta}{2}}WL_{0}^{\frac{-1-\theta}{2}}&L_{0}^{\frac{\theta}{2}}\alpha^{2}L_{0}^{-\frac{\theta}{2}}\\\
{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&L_{0}^{\frac{-1+\theta}{2}}W^{*}L_{0}^{-\frac{\theta}{2}}\end{bmatrix}.$
(5.26)

(5.26) is bounded for $\theta=\rho$ and $\theta=-\rho$. Hence by interpolation
it is bounded for $-\rho\leq\theta\leq\rho$. Hence $B_{0}$ from
${\mathcal{W}}_{\theta+\frac{1}{2}}$ to ${\mathcal{W}}_{\theta-\frac{1}{2}}$.

(5.26) can be represented as

$\displaystyle\begin{bmatrix}L_{0}^{\frac{\theta}{2}}\alpha
L_{0}^{-\frac{\theta}{2}}&0\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}L_{0}^{\frac{\theta}{2}}\alpha^{-1}WL_{0}^{\frac{-1-\theta}{2}}&{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\\\ {{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&L_{0}^{\frac{-1+\theta}{2}}W^{*}\alpha^{-1}L_{0}^{-\frac{\theta}{2}}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ 0&L_{0}^{\frac{\theta}{2}}\alpha
L_{0}^{-\frac{\theta}{2}}\end{bmatrix}$

The two extreme terms are invertible. Besides,

$\|L_{0}^{\frac{\theta}{2}}\alpha^{-1}WL_{0}^{\frac{-1-\theta}{2}}\|\leq a<1.$

Hence the middle term is also invertible. This proves the invertiility of
$B_{0}(t)$ as a map from ${\mathcal{W}}_{\theta+\frac{1}{2}}$ to
${\mathcal{W}}_{\theta-\frac{1}{2}}$.

Let $-\rho-\frac{1}{2}\leq\theta\leq\rho+\frac{1}{2}$. Then for some
$\theta_{0}\in[-\frac{1}{2},\frac{1}{2}]$ and $n\in\mathbb{Z}$ we have
$\theta=n+\theta_{0}$. Then
${\mathcal{W}}_{\theta,t}=B_{0}(t)^{n}{\mathcal{W}}_{\theta_{0},t}$. But as
Hilbertizable spaces
${\mathcal{W}}_{\theta_{0},t}={\mathcal{W}}_{\theta_{0}}$. But we have just
proved that $B_{0}(t)^{n}$ are bounded invertible in the sense
${\mathcal{W}}_{\theta_{0}}\to{\mathcal{W}}_{\theta_{0}+n}$ for
$\theta_{0}+n|\leq\rho+\frac{1}{2}$. Hence
${\mathcal{W}}_{\theta_{0}+n,t}={\mathcal{W}}_{\theta_{0}+n}$. Hence (2) is
true.

(3) is proven in a similar way as Prop. 5.4. ∎

Note that Assumption 5.5($\rho$) is especially important for
$\rho=\frac{1}{2}$. Then (5.24) holds with $\theta=\frac{1}{2}$, so that
$B(t)$ can be interpreted as an (unbounded) pseudo-unitary generator on the
Krein space ${\mathcal{W}}_{0}$ with the domain ${\mathcal{W}}_{1}$.

###  5.4 The abstract Klein-Gordon operator on a finite interval

Assume that $I=]t_{-},t_{+}[$ is finite. Suppose Assumption 5.1 holds. Let
$R(t,s)$ be the corresponding pseudo-unitary evolution on the Krein space
${\mathcal{W}}_{0}$, whose existence is guaranteed by Prop. 5.4. Thus we are
now in the setting of Subsect. 4.8.

First we define the propagators for the Cauchy data. We start with classical
propagators $E^{\mathrm{PJ}}$, $E^{\vee}$, $E^{\wedge}$.

Suppose next that we are given two pairs of closed subspaces
$({\mathcal{Z}}_{-}^{(-)},{\mathcal{Z}}_{-}^{(+)})$ and
$({\mathcal{Z}}_{+}^{(-)},{\mathcal{Z}}_{+}^{(+)})$ with
${\mathcal{Z}}_{\pm}^{(+)}={\mathcal{Z}}_{\pm}^{(-)Q\perp}$. Assume that each
pair is complementary and they satisfy asymptotic complementarity. (This is
automatic if ${\mathcal{Z}}_{\pm}^{(+)}$ are m-positive and
${\mathcal{Z}}_{\pm}^{(-)}$ are m-negative.) We can now define non-classical
propagators $E^{(+-)}$, $E^{(-+)}$, $E_{\pm}^{(+)}$, $E_{\mp}^{(-)}$. All the
propagators for the Cauchy data are bounded operators on
$L^{2}(I,{\mathcal{W}}_{0})$.

Clearly, we can write

$\displaystyle
E^{\bullet}=\begin{bmatrix}E^{\bullet}_{11}(t)&E^{\bullet}_{12}(t)\\\
E^{\bullet}_{21}(t)&E^{\bullet}_{22}(t)\end{bmatrix}.$ (5.27)

We define the propagators for the abstract Klein-Gordon operator by selecting
the upper right element of the matrix of the propagators for the Cauchy data:

$\displaystyle G^{\bullet}$
$\displaystyle:=E_{12}^{\bullet},\quad\bullet=\mathrm{PJ},\vee,\wedge,(+-),(-+);$
(5.28a) $\displaystyle G_{\pm}^{(+)}$
$\displaystyle:=-\mathrm{i}E_{\pm,12}^{(+)},\quad
G_{\pm}^{(-)}:=-\mathrm{i}E_{\pm,12}^{(-)}.$ (5.28b)

######  Theorem 5.7.

(5.28) are bounded operators on $L^{2}(I,{\mathcal{K}})$. They satisfy

$\displaystyle G^{\mathrm{PJ}*}$ $\displaystyle=-G^{\mathrm{PJ}};$ (5.29a)
$\displaystyle G^{\vee*}$ $\displaystyle=G^{\wedge},\quad G^{(+-)*}=G^{(-+)};$
(5.29b) $\displaystyle G_{\pm}^{(+)*}$ $\displaystyle=G_{\pm}^{(+)},\quad
G_{\pm}^{(-)*}=G_{\pm}^{(-)}.$ (5.29c)

If in addition ${\mathcal{Z}}_{\pm}^{(+)}$ are m-positive and
${\mathcal{Z}}_{\pm}^{(+)}$ are m-negative, then we also have

$G_{\pm}^{(+)}\geq 0,\quad G_{\pm}^{(-)}\geq 0.$ (5.30)

We expect that typically $G^{\bullet}$ with $\bullet=\vee,\wedge,(+-),(-+)$
have a zero nullspace and a dense range. (This will be proven below under some
additional assumptions). If this is the case we can define

$K^{\bullet}:=G^{\bullet-1},\quad\bullet=\vee,\wedge,(+-),(-+).$ (5.31)

######  Proposition 5.8.

Suppose Assumption 5.5($\frac{1}{2}$) holds. Let
$\bullet=\vee,\wedge,(+-),(-+)$.

  1. (1)

Let

$f\in C_{\mathrm{c}}(I,{\mathcal{K}}_{\frac{3}{2}})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{K}}_{\frac{1}{2}}),\quad\alpha^{-2}(-\mathrm{i}\partial_{t}+W)f\in
C_{\mathrm{c}}^{1}(I,{\mathcal{K}}_{-\frac{1}{2}}).$ (5.32)

Then

$G^{\bullet}Kf=f.$ (5.33)

  2. (2)

If the space of $f$ satisfying (5.32) is dense in ${\mathcal{K}}$, then
$\operatorname{{\mathcal{N}}}(G^{\bullet})=\\{0\\}$ and
$\operatorname{{\mathcal{R}}}(G^{\bullet})$ is dense.

  3. (3)

If in addition $I\ni t\mapsto
L(t)^{-\frac{1}{4}}\big{(}\partial_{t}\alpha(t)^{-2}\big{)}L(t)^{-\frac{1}{4}},\quad
L(t)^{-\frac{1}{4}}\big{(}\partial_{t}W(t)\big{)}L(t)^{-\frac{3}{4}}\in
B({\mathcal{K}})$ are continuous, then (5.32) is equivalent to

$f\in C_{\mathrm{c}}(I,{\mathcal{K}}_{\frac{3}{2}})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{K}}_{\frac{1}{2}})\cap
C_{\mathrm{c}}^{2}(I,{\mathcal{K}}_{-\frac{1}{2}}).$ (5.34)

###### Proof.

(1): By Proposition 3.14 (1) we know that if $u\in
C_{\mathrm{c}}(I,{\mathcal{W}}_{1})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{W}}_{0})$, then

$E^{\bullet}\big{(}\partial_{t}+\mathrm{i}B(t)\big{)}u=u.$ (5.35)

$f\in C_{\mathrm{c}}(I,{\mathcal{K}}_{\frac{3}{2}})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{K}}_{\frac{1}{2}})$ implies that
$\alpha^{-2}(-\mathrm{i}\partial_{t}+W)f\in
C_{\mathrm{c}}(I,{\mathcal{K}}_{\frac{1}{2}})$. Therefore,

$u:=\begin{bmatrix}f\\\
-\alpha^{-2}(-\mathrm{i}\partial_{t}+W)f\end{bmatrix}\in
C_{\mathrm{c}}(I,{\mathcal{W}}_{1})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{W}}_{0}).$ (5.36)

Hence we can apply (5.35) to $u$. We have

$\displaystyle Q(-\mathrm{i}\partial_{t}+B)$
$\displaystyle=\begin{bmatrix}L&-\mathrm{i}\partial_{t}+W^{*}\\\
-\mathrm{i}\partial_{t}+W&\alpha^{2}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&(-\mathrm{i}\partial_{t}+W^{*})\alpha^{-2}\\\
0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}-K&0\\\
0&\alpha^{2}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
\alpha^{-2}(-\mathrm{i}\partial_{t}+W)&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(5.37)

For $u$ given by (5.36) we have

$Q(-\mathrm{i}\partial_{t}+B)u=\begin{bmatrix}-Kf\\\ 0\end{bmatrix}.$

We obtain

$\displaystyle
u=E^{\bullet}\big{(}\partial_{t}+\mathrm{i}B(t)\big{)}u=\begin{bmatrix}-E_{12}^{\bullet}Kf\\\
-E_{22}^{\bullet}Kf\end{bmatrix},$ (5.38)

which yields (5.33).

By (5.33) the range of $G^{\bullet}$ contains (5.32) contains. The same
argument shows that the range of $G^{\bullet*}$ contains (5.32). If the space
of (5.32) is dense, we have

$\operatorname{{\mathcal{N}}}(G^{\bullet})=\operatorname{{\mathcal{R}}}(G^{\bullet*})^{\perp}=\\{0\\}.$
(5.39)

This proves (2).

(3) follows by checking that $\partial_{t}\alpha^{-2}\partial_{t}f$ and
$\partial_{t}\alpha^{-2}Wf$ are in
$C_{\mathrm{c}}(I,{\mathcal{K}}_{-\frac{1}{2}})$. ∎

Thus, under the assumptions of Prop. 5.8 (3) we can define an unbounded closed
operator $K^{\bullet}$ in the sense of $L^{2}(I,{\mathcal{K}})$ with $0$
belonging to its resolvent set. It corresonds to the quadratic form (5.5) with
the appropriate boundary conditions. For instance, the boundary conditions of
$K^{(+-)}$ are

$\displaystyle f\in{\mathcal{D}}(K^{(+-)})$
$\displaystyle\Rightarrow\quad\begin{bmatrix}f(t_{\mp})\\\
-\alpha^{-2}(-\mathrm{i}\partial_{t}+W)f(t_{\mp})\end{bmatrix}\in{\mathcal{Z}}_{\mp}^{(\mp)},$
$\displaystyle f\in{\mathcal{D}}(K^{{(-+)}})$
$\displaystyle\Rightarrow\quad\begin{bmatrix}f(t_{\mp})\\\
-\alpha^{-2}(-\mathrm{i}\partial_{t}+W)f(t_{\mp})\end{bmatrix}\in{\mathcal{Z}}_{\mp}^{(\pm)}.$

In particular, this construction works if ${\mathcal{Z}}_{\pm}^{(+)}$ are
m-positive subspaces of ${\mathcal{W}}_{0}$. Then ${\mathcal{Z}}_{\pm}^{(-)}$
are m-negative subspaces. We obtain a pair of operators
$(K^{(+-)})^{*}=K^{(-+)}$ whose resolvent sets contain $\mathbb{R}$.

Clearly, the above construction is indirect. It does not mean that the
expression (5.1) is well defined in the sense of the Hilbert space
$L^{2}(I,{\mathcal{K}})$. Under more stringent conditions, such as those
described in the following proposition, one can directly interpret (5.1) as an
operator:

######  Proposition 5.9.

Suppose that Assumption 5.5($1$) holds. In addition, assume that
$I^{\mathrm{cl}}\ni
t\mapsto\big{(}\partial_{t}\alpha^{-2}(t)\big{)}L_{0}(t)^{-\frac{1}{2}}$,
$\big{(}\partial_{t}W(t)\big{)}L_{0}(t)^{-1}\in B({\mathcal{K}})$ are norm
continuous families. Then $K$, as defined in (5.1), maps

$C_{\mathrm{c}}(I,{\mathcal{K}}_{2})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{K}}_{1})\cap
C_{\mathrm{c}}^{2}(I,{\mathcal{K}}_{0})$ (5.40)

into $C_{\mathrm{c}}(I,{\mathcal{K}}_{0})$. Hence (5.40) can serve as a dense
domain of the operator $K$.

###### Proof.

We rewrite $K$ as

$\displaystyle K$
$\displaystyle=\frac{1}{\alpha^{2}(t)}D_{t}^{2}+W^{*}(t)\frac{1}{\alpha^{2}(t)}D_{t}+\frac{1}{\alpha^{2}(t)}W(t)D_{t}+\Big{(}\partial_{t}\frac{\mathrm{i}}{\alpha^{2}(t)}\Big{)}D_{t}$
$\displaystyle+W^{*}(t)\frac{1}{\alpha^{2}(t)}W(t)+\Big{(}\partial_{t}\frac{\mathrm{i}}{\alpha^{2}(t)}\Big{)}W(t)-\frac{\mathrm{i}}{\alpha^{2}(t)}\partial_{t}W(t)-L(t).$

∎

###  5.5 The abstract Klein-Gordon operator on the real line

We can try to repeat the analysis of the previous subsection for
$I=\mathbb{R}$. Assuming that Assumption 5.1 holds, we can define the
evolution $R(t,s)$ on the Krein space ${\mathcal{W}}_{0}$ and define the
operators $E^{\mathrm{PJ}}$, $E^{\vee}$, $E^{\wedge}$. After choosing families
of pairs of spaces
$\big{(}{\mathcal{Z}}_{\pm}^{(+)}(t),{\mathcal{Z}}_{\pm}^{(-)}(t)\big{)}$
satisfying (3.18) and (4.44), which we recall below

$\displaystyle
R(t,s){\mathcal{Z}}_{\pm}^{(+)}(s)={\mathcal{Z}}_{\pm}^{(+)}(t),$
$\displaystyle\quad
R(t,s){\mathcal{Z}}_{\pm}^{(-)}(s)={\mathcal{Z}}_{\pm}^{(-)}(t),$ (5.41)
$\displaystyle{\mathcal{Z}}_{-}^{(+)}(t)={\mathcal{Z}}_{-}^{(-)}(t)^{\perp
Q},$
$\displaystyle\quad{\mathcal{Z}}_{+}^{(-)}(t)={\mathcal{Z}}_{+}^{(+)}(t)^{\perp
Q}.$ (5.42)

Assuming m-positivity of ${\mathcal{Z}}_{\pm}^{(+)}(t)$ (which implies
asymptotic complementarity), we can define $E_{\pm}^{(+)}$, $E_{\pm}^{(-)}$,
$E^{(+-)}$, $E^{(-+)}$, understood as operators from
$L_{\mathrm{c}}^{2}(\mathbb{R},{\mathcal{W}}_{0})$ to
$L_{\mathrm{loc}}^{2}(\mathbb{R},{\mathcal{W}}_{0})$. We can define the
operators $G^{\bullet}$, $G_{\pm}^{(+)}$, $G_{\pm}^{(-)}$ just as in (5.28),
interpreted as operators $L_{\mathrm{c}}^{2}(\mathbb{R},{\mathcal{K}})$ to
$L_{\mathrm{loc}}^{2}(\mathbb{R},{\mathcal{K}})$. Obvious analogs of Prop. 5.8
and 5.9 hold for $I=\mathbb{R}$.

There is however a big difference with the case of a finite interval $I$: the
above operators are not bounded and $G^{\bullet-1}$,
$\bullet=\vee,\wedge,(+-),(-+),$ do not define closed realizations of the
Klein-Gordon operator. Actually, in the above generality, they are probably
not very interesting both from the physical and mathematical point of view.
Physically, the spaces ${\mathcal{Z}}_{\pm}^{(+)}$,
${\mathcal{Z}}_{\pm}^{(-)}$ do not usually correspond to realistic Fock
representations. Mathematically, the corresponding boundary conditions do not
define closed reaizations of the Klein-Gordon operator.

There is in practice one choice of
$\big{(}{\mathcal{Z}}_{\pm}^{(+)}(t),{\mathcal{Z}}_{\pm}^{(+)}(t)\big{)}$
which is important, both physically and mathematically, which we describe
below. This choice should work if $B(t)$ is asymptotically stationary.
However, for simplicity, in order to describe this choice let us make a
stronger assumption:

$B(t)$ does not depend on $t$ for $|t|>T$, (5.43)

so that there exist $B(\pm\infty):=\lim\limits_{t\to\pm\infty}B(t)$. We also
assume that

$B(\pm\infty)\quad\text{are stable (see Def. \ref{def:1}).}$ (5.44)

We set

$\displaystyle{\mathcal{Z}}_{\pm}^{(+)}(t)$
$\displaystyle=\operatorname{{\mathcal{R}}}\Big{(}\lim_{s\pm\infty}R(t,s){{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}_{[0,\infty[}\big{(}B(\pm\infty)\big{)}R(s,t)\Big{)},$
(5.45a) $\displaystyle{\mathcal{Z}}_{\pm}^{(-)}(t)$
$\displaystyle=\operatorname{{\mathcal{R}}}\Big{(}\lim_{s\pm\infty}R(t,s){{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}_{]-\infty,0]}\big{(}B(\pm\infty)\big{)}R(s,t)\Big{)}.$
(5.45b)

Clearly, ${\mathcal{Z}}_{\pm}^{(+)}(t)$ are m-positive and
${\mathcal{Z}}_{\pm}^{(-)}(t)$ are m-negative, and hence the asymptotic
complementarity holds.

######  Definition 5.10.

With the choice (5.45), the bisolutions $G_{\pm}^{(+)}$ are called the _in/out
positive frequency bisolutions_ , the bisolutions $G_{\pm}^{(-)}$ are called
the _in/out negative frequency bisolutions_ , and the inverses $G^{(+-)}$,
resp. $G^{(-+)}$ are called the _Feynman_ , resp. the _anti-Feynman inverse_.

It is clear that the above propagators play a central role in Quantum Field
Theory. In fact, they correspond to positive energy Fock representations of
incoming and outgoing quantum fields, which is viewed as a reasonable
assumption on the physical grounds.

We actually believe that this choice is also distinguished for very different
reasons by purely mathematical arguments. It probably corresponds to a
distinguished (maybe unique) self-adjoint realization of the Klein-Gordon
operator. We formulate our expectation in the following conjecture.

######  Conjecture 5.11.

For a large class of asymptotically stationary and stable abstract Klein-
Gordon operators (that is, satisfying Assumption 5.1, (5.43) and (5.44) and
perhaps some other conditions) the following holds:

  1. (1)

There exists a distinguished self-adjoint operator $K^{\mathrm{s.a.}}$ on
$L^{2}(\mathbb{R},{\mathcal{K}})$ such that the quadratic form (5.5) coincides
with $(f_{1}|K^{\mathrm{s.a.}}f_{2})$.

  2. (2)

In the sense of $\langle
t\rangle^{-s}L^{2}({{\mathbb{R}}},{\mathcal{K}})\to\langle
t\rangle^{s}L^{2}({{\mathbb{R}}},{\mathcal{K}})$, for $s>\frac{1}{2}$,

$\displaystyle\operatorname*{s-lim}_{\epsilon\searrow
0}(K^{\mathrm{s.a.}}-\mathrm{i}\epsilon)^{-1}$ $\displaystyle=G^{{(+-)}},$
(5.46) $\displaystyle\operatorname*{s-lim}_{\epsilon\searrow
0}(K^{\mathrm{s.a.}}+\mathrm{i}\epsilon)^{-1}$ $\displaystyle=G^{{(-+)}}.$
(5.47)

  3. (3)

The operator $K$ with the domain, say,
$C_{\mathrm{c}}^{2}(\mathbb{R},{\mathcal{K}}_{0})\cap
C_{\mathrm{c}}^{1}(\mathbb{R},{\mathcal{K}}_{1})\cap
C_{\mathrm{c}}(\mathbb{R},{\mathcal{K}}_{2})$ is essentially self-adjoint and
its closure is $K^{\mathrm{s.a.}}$.

We will discuss arguments in favor of this conjecture in Subsect. 5.8.

###  5.6 Perturbation of the evolution by the spectral parameter

In the remaining part of this section we would like to describe how to compute
the resolvent of well-posed realizations of the Klein-Gordon operator. To this
end in this subsection we introduce the perturbed evolution $R_{z}(t,s)$.

Let $z\in\mathbb{C}$. We define

$\displaystyle Z$ $\displaystyle\coloneqq\begin{bmatrix}0&0\\\
{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\end{bmatrix},$ (5.48) $\displaystyle B_{z}(t)$
$\displaystyle\coloneqq\begin{bmatrix}W(t)&\alpha^{2}(t)\\\
L(t)-z&W^{*}(t)\end{bmatrix}=B(t)-zZ.$ (5.49)

Note that $Z$ is a bounded operator on each ${\mathcal{W}}_{\lambda}$:

$\|Zv\|_{\lambda}=\|v_{1}\|_{\lambda-\frac{1}{2}}\leq\|v_{1}\|_{\lambda+\frac{1}{2}}+\|v_{2}\|_{\lambda-\frac{1}{2}}=\|v\|_{\lambda}.$

######  Proposition 5.12.

Suppose Assumption 5.1 holds. Then $t\mapsto B_{z}(t)$ generates an evolution
$R_{z}(t,s)$ on ${\mathcal{W}}_{\lambda}$,
$\lambda\in[-\frac{1}{2},\frac{1}{2}]$. We have

$R_{\overline{z}}(t,s)^{*}QR_{z}(t,s)=Q,\quad
R_{z}(t,s)^{*}=QR_{\overline{z}}(s,t)Q.$ (5.50)

In particular, if $z\in\mathbb{R}$, then $R_{z}(t,s)$ is pseudo-unitary on
${\mathcal{W}}_{0}$.

Formally the equation

$(z+K)f=0$ (5.51)

is equivalent to

$\displaystyle\mathrm{i}\partial_{t}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}$ $\displaystyle=B_{z}(t)\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix},$ (5.52) $\displaystyle u_{1}=f,$ $\displaystyle\quad
u_{2}=\frac{1}{\alpha^{2}(t)}\big{(}D_{t}+W(t)\big{)}f.$ (5.53)

Therefore, the evolution $R_{z}(t,s)$ can be used to construct the resolvent
of realizations of $K$.

###  5.7 Resolvent for finite intervals

Assume again that $I$ is finite. We are back in the setting of Subsect. 5.4.
Recall that we impose Assumption 5.1 and choose
$({\mathcal{Z}}_{-}^{(-)},{\mathcal{Z}}_{-}^{(+)})$ and
$({\mathcal{Z}}_{+}^{(-)},{\mathcal{Z}}_{+}^{(+)})$ with
${\mathcal{Z}}_{\pm}^{(+)}={\mathcal{Z}}_{\pm}^{(-)Q\perp}$. For
$z\in{{\mathbb{C}}}$ and $t\in I$, set

${\mathcal{Z}}_{\mp,z}^{(-)}(t):=R_{z}(t,t_{\mp}){\mathcal{Z}}_{\mp}^{(-)},\quad{\mathcal{Z}}_{\pm,z}^{(+)}(t):=R_{z}(t,t_{\pm}){\mathcal{Z}}_{+}^{(+)}.$
(5.54)

Let

$\displaystyle\mathrm{RS}:=\\{z\in{{\mathbb{C}}}\mid$ for some (hence all)
$t\in I$ the pair of subspaces
$\displaystyle\text{$\big{(}{\mathcal{Z}}_{+,z}^{(+)}(t),{\mathcal{Z}}_{-,z}^{(-)}(t)\big{)}$
is complementary}\\}.$

By Prop. 4.11 (2),

$\displaystyle\mathrm{RS}:=\\{z\in{{\mathbb{C}}}\mid$ for some (hence all)
$t\in I$ the pair of subspaces
$\displaystyle\text{$\big{(}{\mathcal{Z}}_{+,z}^{(-)}(t),{\mathcal{Z}}_{-,z}^{(+)}(t)\big{)}$
is complementary}\\}.$

For $z\in\mathrm{RS}$ we define

$\displaystyle\Lambda_{z}^{(\boxplus-)}(t),$ $\displaystyle\quad\text{ the
projection onto ${\mathcal{Z}}_{+,z}^{(+)}(t)$ along
${\mathcal{Z}}_{-,z}^{(-)}(t)$},$
$\displaystyle\Lambda_{z}^{(+\boxminus)}(t),$ $\displaystyle\quad\text{ the
projection onto ${\mathcal{Z}}_{-,z}^{(-)}(t)$ along
${\mathcal{Z}}_{+,z}^{(+)}(t)$};$ $\displaystyle\Lambda_{z}^{(-\boxplus)}(t),$
$\displaystyle\quad\text{ the projection onto ${\mathcal{Z}}_{-,z}^{(+)}(t)$
along ${\mathcal{Z}}_{+,z}^{(-)}(t)$},$
$\displaystyle\Lambda_{z}^{(\boxminus+)}(t),$ $\displaystyle\quad\text{ the
projection onto ${\mathcal{Z}}_{+,z}^{(-)}(t)$ along
${\mathcal{Z}}_{-,z}^{(+)}(t)$}.$

Set

$\displaystyle E_{z}^{(+-)}(t,s)$
$\displaystyle\coloneqq\theta(t-s)R_{z}(t,s)\Lambda_{z}^{(\boxplus-)}(s)-\theta(s-t)R_{z}(t,s)\Lambda_{z}^{(+\boxminus)}(s),$
(5.55) $\displaystyle E_{z}^{(-+)}(t,s)$
$\displaystyle\coloneqq\theta(t-s)R_{z}(t,s)\Lambda_{z}^{(\boxminus+)}(s)-\theta(s-t)R_{z}(t,s)\Lambda_{z}^{(-\boxplus)}(s);$
(5.56) $\displaystyle G_{z}^{(+-)}(t,s)$
$\displaystyle\coloneqq-\mathrm{i}E_{z,12}^{(+-)}(t,s),$ (5.57) $\displaystyle
G_{z}^{(-+)}(t,s)$ $\displaystyle\coloneqq-\mathrm{i}E_{z,12}^{(-+)}(t,s).$
(5.58)

######  Proposition 5.13.

For $z\in\mathrm{RS}$ the operators $G_{z}^{(+-)}$, $G_{\overline{z}}^{(-+)}$
are bounded and satisfy the resolvent equation:

$\displaystyle G_{z}^{(+-)}-G_{w}^{(+-)}$
$\displaystyle=(z-w)G_{z}^{(+-)}G_{w}^{(+-)},$ (5.59) $\displaystyle
G_{z}^{(-+)}-G_{w}^{(-+)}$ $\displaystyle=(z-w)G_{z}^{(-+)}G_{w}^{(-+)}.$
(5.60)

Besides,

$\displaystyle(QE_{z}^{(+-)})^{*}$ $\displaystyle=-QE_{\overline{z}}^{(-+)},$
(5.61) $\displaystyle(G_{z}^{(+-)})^{*}$
$\displaystyle=G_{\overline{z}}^{(-+)}.$ (5.62)

###### Proof.

The boundedness is obvious. Let us prove (5.59).

$\displaystyle(z-w)E_{z}^{(+-)}ZE_{w}^{(+-)}(t,s)$ $\displaystyle=\quad$
$\displaystyle\theta(t-s)(z-w)\int_{s}^{t}\Lambda_{z}^{(\boxplus-)}(t)R_{z}(t,\tau)ZR_{w}(\tau,s)\Lambda_{w}^{(\boxplus-)}(s)\mathop{}\\!\mathrm{d}\tau$
$\displaystyle+$
$\displaystyle\theta(s-t)(z-w)\int_{t}^{s}\Lambda_{z}^{(+\boxminus)}(t)R_{z}(t,\tau)ZR_{w}(\tau,s)\Lambda_{w}^{(+\boxminus)}(s)\mathop{}\\!\mathrm{d}\tau$
$\displaystyle-$
$\displaystyle\theta(t-s)(z-w)\int_{t_{-}}^{s}\Lambda_{z}^{(\boxplus-)}(t)R_{z}(t,\tau)ZR_{w}(\tau,s)\Lambda_{w}^{(+\boxminus)}(s)\mathop{}\\!\mathrm{d}\tau$
$\displaystyle-$
$\displaystyle\theta(s-t)(z-w)\int_{t_{-}}^{t}\Lambda_{z}^{(\boxplus-)}(t)R_{z}(t,\tau)ZR_{w}(\tau,s)\Lambda_{w}^{(+\boxminus)}(s)\mathop{}\\!\mathrm{d}\tau$
$\displaystyle-$
$\displaystyle\theta(s-t)(z-w)\int_{s}^{t_{+}}\Lambda_{z}^{(+\boxminus)}(t)R_{z}(t,\tau)ZR_{w}(\tau,s)\Lambda_{w}^{(\boxplus-)}(s)\mathop{}\\!\mathrm{d}\tau$
$\displaystyle-$
$\displaystyle\theta(t-s)(z-w)\int_{t}^{t_{+}}\Lambda_{z}^{(+\boxminus)}(t)R_{z}(t,\tau)ZR_{w}(\tau,s)\Lambda_{w}^{(\boxplus-)}(s)\mathop{}\\!\mathrm{d}\tau.$

This yields

$\displaystyle=\quad$
$\displaystyle\theta(t-s)\Lambda_{z}^{(\boxplus-)}(t)\Big{(}R_{z}(t,s)-R_{w}(t,s)\Big{)}\Lambda_{w}^{(\boxplus-)}(s)$
$\displaystyle+$
$\displaystyle\theta(s-t)\Lambda_{z}^{(+\boxminus)}(t)\Big{(}R_{w}(t,s)-R_{z}(t,s)\Big{)}\Lambda_{w}^{(+\boxminus)}(s)$
$\displaystyle-$
$\displaystyle\theta(t-s)\Lambda_{z}^{(\boxplus-)}(t)\Big{(}R_{z}(t,t_{-})R_{w}(t_{-},s)-R_{z}(t,s)\Big{)}\Lambda_{w}^{(+\boxminus)}(s)$
$\displaystyle-$
$\displaystyle\theta(s-t)\Lambda_{z}^{(\boxplus-)}(t)\Big{(}R_{z}(t,t_{-})R_{w}(t_{-},s)-R_{w}(t,s)\Big{)}\Lambda_{w}^{(+\boxminus)}(s)$
$\displaystyle-$
$\displaystyle\theta(s-t)\Lambda_{z}^{(+\boxminus)}(t)\Big{(}-R_{z}(t,t_{+})R_{w}(t_{+},s)+R_{z}(t,s)\Big{)}\Lambda_{w}^{(\boxplus-)}(s)$
$\displaystyle-$
$\displaystyle\theta(t-s)\Lambda_{z}^{(+\boxminus)}(t)\Big{(}-R_{z}(t,t_{+})R_{w}(t_{+},s)+R_{w}(t,s)\Big{)}\Lambda_{w}^{(\boxplus-)}(s).$

We rearrange this, obtaining

$\displaystyle=\quad$
$\displaystyle\theta(t-s)\Lambda_{z}^{(\boxplus-)}(t)R_{z}(t,s)\Big{(}\Lambda_{w}^{(\boxplus-)}(s)+\Lambda_{w}^{(+\boxminus)}(s)\Big{)}$
$\displaystyle-$
$\displaystyle\theta(t-s)\Big{(}\Lambda_{z}^{(+\boxminus)}(t)+\Lambda_{z}^{(\boxplus-)}(t)\Big{)}R_{w}(t,s)\Lambda_{w}^{(\boxplus-)}(s)$
$\displaystyle-$
$\displaystyle\theta(s-t)\Lambda_{z}^{(+\boxminus)}(t)R_{z}(t,s)\Big{(}\Lambda^{(\boxplus-)}_{w}(s)+\Lambda_{w}^{(+\boxminus)}(s)\Big{)}$
$\displaystyle+$
$\displaystyle\theta(s-t)\Big{(}\Lambda_{z}^{(+\boxminus)}(t)+\Lambda_{z}^{(\boxplus-)}(t)\Big{)}R_{w}(t,s)\Lambda_{w}^{(+\boxminus)}(s)$
$\displaystyle-$ $\displaystyle
R_{z}(t,t_{-})\Lambda_{z}^{(\boxplus-)}(t_{-})\Lambda_{w}^{(+\boxminus)}(t_{-})R_{w}(t_{-},s)$
(5.63) $\displaystyle+$ $\displaystyle
R_{z}(t,t_{+})\Lambda_{z}^{(+\boxminus)}(t_{+})\Lambda_{w}^{(\boxplus-)}(t_{+})R_{w}(t_{+},s).$
(5.64)

Now, for any $z,w$,

$\displaystyle\operatorname{{\mathcal{R}}}\big{(}\Lambda_{w}^{(+\boxminus)}(t_{-})\big{)}$
$\displaystyle={\mathcal{Z}}_{-}^{(-)}=\operatorname{{\mathcal{N}}}\big{(}\Lambda_{z}^{(\boxplus-)}(t_{-})\big{)}$
$\displaystyle\operatorname{{\mathcal{R}}}\big{(}\Lambda_{w}^{(\boxplus-)}(t_{+})\big{)}$
$\displaystyle={\mathcal{Z}}_{+}^{(+)}=\operatorname{{\mathcal{N}}}\big{(}\Lambda_{z}^{(+\boxminus)}(t_{+})\big{)}.$

Therefore, the terms are (5.63) and (5.64) are zero. Therefore, we obtain

$\displaystyle=\quad$
$\displaystyle\theta(t-s)\Lambda_{z}^{(\boxplus-)}(t)R_{z}(t,s)-\theta(t-s)R_{w}(t,s)\Lambda_{w}^{(\boxplus-)}(s)$
$\displaystyle-$
$\displaystyle\theta(s-t)\Lambda_{z}^{(+\boxminus)}(t)R_{z}(t,s)+\theta(s-t)R_{w}(t,s)\Lambda_{w}^{(+\boxminus)}(s)$
$\displaystyle=\quad$ $\displaystyle E_{z}^{(+-)}(t,s)-E_{w}^{(+-)}(t,s).$

Thus we have proven that

$(z-w)E_{z}^{(+-)}ZE_{w}^{(+-)}(t,s)=E_{z}^{(+-)}(t,s)-E_{w}^{(+-)}(t,s).$
(5.65)

Taking the $1,2$ component of (7.5) we obtain (5.59).

To prove (5.61) we use (5.50). This implies (5.62). ∎

Clearly, if we can define $K^{(+-)}$, $K^{(-+)}$ by (5.31) as operators with
dense domains, then we have

$G_{z}^{(+-)}=(z+K^{(+-)})^{-1},\quad G_{z}^{(-+)}=(z+K^{(-+)})^{-1}.$ (5.66)
resolvent set of $K^{(+-)}$ $\displaystyle\,=\,\overline{\text{resolvent set
of $K^{{(-+)}}$}}\,\subset\,\mathrm{RS}.$

###  5.8 Resolvent for $I=\mathbb{R}$

In this subsection we will give some arguments supporting Conjecture 8.3. We
will sketch a construction of a family of operators which we expect is the
resolvent of the (putatative) distinguished self-adjoint realization of the
abstract Klein-Gordon operator for $I=\mathbb{R}$. We impose Assumption 5.1
and the condition (5.43) (but it is likely that more assumptions are needed).
Our analysis will not be complete.

First note that we cannot repeat the constructions of Subsection 5.7 without
major changes. In fact, for $I=\mathbb{R}$ in the definitions (5.54) one
should take $t\to\pm\infty$. However, for $z\not\in\mathbb{R}$ the evolution
$R_{z}(t,s)$ blows up on a part of ${\mathcal{W}}_{0}$ and decays on another
part as $t\to\pm\infty$.

Before we proceed let us apply Proposition 7.2 of [6] to $B(\pm\infty)$. For
simplicity, we will assume that $B(\pm\infty)$ are strongly stable, so that
their spectrum has a gap around $0$.

######  Proposition 5.14.

There exists $\zeta_{0}>0$ such that the strip $\\{\zeta\in\mathbb{C}\ |\
-\zeta_{0}\leq\operatorname{Re}(\zeta)\leq\zeta_{0}\\}$ is contained in the
resolvent set of $B_{z}(\pm\infty)$. Moreover, the operators

$\displaystyle\Pi_{\pm,z}^{(+)}$
$\displaystyle:=\lim_{\tau\to\infty}\frac{1}{2}\Big{(}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+\frac{1}{\pi\mathrm{i}}\int_{-\mathrm{i}\tau}^{\mathrm{i}\tau}(B_{z}(\pm\infty)-\zeta)^{-1}\mathop{}\\!\mathrm{d}\zeta\Big{)},$
(5.67) $\displaystyle\Pi_{\pm,z}^{(-)}$
$\displaystyle:=\lim_{\tau\to\infty}\frac{1}{2}\Big{(}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-\frac{1}{\pi\mathrm{i}}\int_{-\mathrm{i}\tau}^{\mathrm{i}\tau}(B_{z}(\pm\infty)-\zeta)^{-1}\mathop{}\\!\mathrm{d}\zeta\Big{)}$
(5.68)

constitute a pair of complementary projections commuting with
$B_{z}(\pm\infty)$ such that

$\displaystyle\sigma\big{(}B_{z}(\pm\infty)\Pi_{\pm,z}^{(+)}\big{)}$
$\displaystyle=\sigma\big{(}B_{z}(\pm\infty)\big{)}\cap\\{w\in\mathbb{C}\ |\
\operatorname{Re}(w)\geq 0\\},$ (5.69)
$\displaystyle\sigma\big{(}B_{z}(\pm\infty)\Pi_{\pm,z}^{(-)}\big{)}$
$\displaystyle=\sigma\big{(}B_{z}(\pm\infty)\big{)}\cap\\{w\in\mathbb{C}\ |\
\operatorname{Re}(w)\leq 0\\}.$ (5.70)

Set

$\displaystyle{\mathcal{Z}}_{\pm,z}^{(\pm)}(t)$
$\displaystyle:=\operatorname{{\mathcal{R}}}\Big{(}\lim_{\tau\to\pm\infty}R_{z}(t,\tau)\Pi_{\pm,z}^{(\pm)}R_{z}(\tau,t)\Big{)},\quad\operatorname{Im}(z)\geq
0$ (5.71) $\displaystyle{\mathcal{Z}}_{\pm,z}^{(\mp)}(t)$
$\displaystyle:=\operatorname{{\mathcal{R}}}\Big{(}\lim_{\tau\to\pm\infty}R_{z}(t,\tau)\Pi_{\pm,z}^{(\mp)}R_{z}(\tau,t)\Big{)},\quad\operatorname{Im}(z)\leq
0.$ (5.72)

We can now complement Conjecture 8.3 with an additional conjecture about the
resolvent of $K^{\mathrm{s.a.}}$:

######  Conjecture 5.15.

We expect that for $\operatorname{Re}(z)\geq 0$ for some (hence all)
$t\in\mathbb{R}$ the pair of subspaces
$\big{(}{\mathcal{Z}}_{+,z}^{(+)}(t),{\mathcal{Z}}_{-,z}^{(-)}(t)\big{)}$ is
complementary. Let
$\big{(}\Lambda_{z}^{(+\boxminus)}(t),\Lambda_{z}^{(\boxplus-)}(t)\big{)}$ be
the pair of projections corresponding to this pair of spaces.

Equivalently, we expect that for $\operatorname{Re}(z)\leq 0$ for some (hence
all) $t\in\mathbb{R}$ the pair of subspaces
$\big{(}{\mathcal{Z}}_{+,z}^{(-)}(t),{\mathcal{Z}}_{-,z}^{(+)}(t)\big{)}$ is
complementary. Let
$\big{(}\Lambda_{z}^{(\boxminus+)}(t),\Lambda_{z}^{(-\boxplus)}(t)\big{)}$ be
the pair of projections corresponding to this pair of spaces.

We also introduce

$\displaystyle E_{z}(t,s)$
$\displaystyle\coloneqq\theta(t-s)R_{z}(t,s)\Lambda_{z}^{(\boxplus-)}(s)-\theta(s-t)R_{z}(t,s)\Lambda_{z}^{(+\boxminus)}(s),\quad\operatorname{Re}(z)>0;$
(5.73) $\displaystyle E_{z}(t,s)$
$\displaystyle\coloneqq\theta(t-s)R_{z}(t,s)\Lambda_{z}^{(\boxminus+)}(s)-\theta(s-t)R_{z}(t,s)\Lambda_{z}^{(-\boxplus)}(s),\quad\operatorname{Re}(z)<0;$
(5.74) $\displaystyle G_{z}(t,s)$
$\displaystyle\coloneqq-\mathrm{i}E_{z,12}(t,s),\quad\operatorname{Re}(z)\neq
0.$ (5.75)

Then we conjecture that $G_{z}$ defines for
$z\in\mathbb{C}\backslash\mathbb{R}$ a bounded operator on
$L^{2}(\mathbb{R},{\mathcal{K}})$ with a dense range and a trivial null-space
such that

$\displaystyle G_{z}-G_{w}$ $\displaystyle=(z-w)G_{z}G_{w},$ (5.76)
$\displaystyle G_{z}^{*}$ $\displaystyle=G_{\overline{z}}.$ (5.77)

Thus $G_{z}$ is a resolvent of a self-adjoint operator, which we can call
$K^{\mathrm{s.a}}$ and treat as the distinguished self-adjoint realization of
$K$.

Let us sketch some arguments in favor of the above conjecture.

First of all, in the stationary case, that is if the $B(t)$ does not depend on
$t$, then the conjecture is true (with minor additional assumptions) following
the arguments of [6] and [7].

The conjecture is also true if ${\mathcal{K}}=\mathbb{C}$. In fact, the
operator $K$ is then essentially the well-known 1-dimensional magnetic
Schrödinger operator. It should not be difficult to generalize this to the
case of a finite dimensional ${\mathcal{K}}$, or bounded $L(t)$ and $W(t)$.

Unfortunately, for a generic spacetime, $L(t)$ is unbounded and non-
stationary. We know in this case that asymptotic complementarity holds for
real $z$, so we can expect it to hold in a neighborhood of $\mathbb{R}$.
$E_{z}$ are well defined as quadratic forms on, say,
$C_{\mathrm{c}}(\mathbb{R},{\mathcal{W}}_{0})$ and (5.77) is easily checked.
However, we do not know how to control the norm of $E_{z}(t,s)$ for large
$t,s$, and hence to show the boundedness of $E_{z}$.

We expect that

$\displaystyle\lim_{t\to+\infty}\mathrm{e}^{\mp\mathrm{i}tB_{z}(\pm\infty)}\Pi_{\pm,z}^{(\pm)}$
$\displaystyle=0,\quad\operatorname{Im}z\geq 0;$ (5.78)
$\displaystyle\lim_{t\to-\infty}\mathrm{e}^{\mp\mathrm{i}tB_{z}(\pm\infty)}\Pi_{\pm,z}^{(\mp)}$
$\displaystyle=0,\quad\operatorname{Im}z\leq 0.$ (5.79)

(This follows under a sligtly stronger assumption from Proposition 7.2 of
[6]). Now the resolvent equation (5.76) should follow by the same calculation
as in the proof of Proposition 5.13, except that the terms (5.63) and (5.64)
should be zero by (5.78) and (5.79).

##  6 Bosonic quantization: real (or neutral) formalism

In this and the next section we briefly describe the well-known neutral
formalism of bosonic quantization, following mostly [5]. The complex formalism
will be introduced in the next section.

###  6.1 Canonical commutation relations

Suppose that ${\mathcal{Y}}$ is a real vector space equipped with an
antisymmetric form $\omega$, i.e., $({\mathcal{Y}},\omega)$ is a pre-
symplectic space.

Let $\mathrm{CCR}({\mathcal{Y}})$ denote the complex unital $*$-algebra
generated by $\phi(w)$, $w\in{\mathcal{Y}}$, satisfying

  1. (1)

$\phi(w)^{*}=\phi(w)$,

  2. (2)

the map ${\mathcal{Y}}\ni w\mapsto\phi(w)$ is linear,

  3. (3)

and the canonical commutation relations hold,

$\bigl{[}\phi(v),\phi(w)\bigr{]}=\mathrm{i}\langle v\mkern 2.0mu|\mkern
2.0mu\mathopen{}\omega w\rangle,\quad v,w\in{\mathcal{Y}}.$ (6.1)

Let ${\mathcal{W}}\coloneqq{{\mathbb{C}}}{\mathcal{Y}}$ be the
complexification of ${\mathcal{Y}}$ and $Q$ the corresponding Hermitian form,
as described in (2.11). We extend $\phi$ to ${\mathcal{W}}$, so that it is
complex antilinear:

$\phi(w_{R}+\mathrm{i}w_{I})\coloneqq\phi(w_{R})-\mathrm{i}\phi(w_{I}),\quad
w_{R},w_{I}\in{\mathcal{Y}}.$

Then we have, for all $v,w\in{\mathcal{W}}$,

$\displaystyle\phi^{*}(w)\coloneqq\phi(w)^{*}$
$\displaystyle=\phi(\overline{w}),$
$\displaystyle\bigl{[}\phi(v),\phi^{*}(w)\bigr{]}$ $\displaystyle=(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}Qw).$

###  6.2 Fock representation

Assume in addition that ${\mathcal{W}}$ is Krein. Let $S_{\bullet}$ be an
admissible anti-real involution on ${\mathcal{W}}$, see Subsect. 4.2. Let
${\mathcal{Z}}_{\bullet}^{(+)}$, $\Pi_{\bullet}^{(+)}$ be the corresponding
particle space and projection, see Subsect. 2.8.

It is well-known that the two-point function

$\omega_{\bullet}\bigl{(}\phi(v)\phi^{*}(w)\bigr{)}=(\Pi_{\bullet}^{(+)}v\mkern
2.0mu|\mkern 2.0mu\mathopen{}Q\Pi_{\bullet}^{(+)}w),\quad
v,w\in{\mathcal{W}}.$

uniquely determines a centered pure quasi-free state on the algebra
$\mathrm{CCR}({\mathcal{Y}})$. The corresponding GNS representation acts on
the bosonic Fock space $\Gamma_{\mathrm{s}}({\mathcal{Z}}_{\bullet}^{(+)})$
and the state is given by the vacuum
$\Omega_{\bullet}\in\Gamma_{\mathrm{s}}({\mathcal{Z}}_{\bullet}^{(+)})$:

$\omega_{\bullet}(\,\cdot\,)=(\Omega_{\bullet}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\,\cdot\;\Omega_{\bullet}).$

The representation is given by

$\displaystyle\phi_{\bullet}(w)$ $\displaystyle\coloneqq
a_{\bullet}(\Pi_{\bullet}^{(+)}w)+a_{\bullet}^{*}(\Pi_{\bullet}^{(+)}\overline{w}),$
$\displaystyle\phi_{\bullet}^{*}(w)$ $\displaystyle\coloneqq
a_{\bullet}^{*}(\Pi_{\bullet}^{(+)}w)+a_{\bullet}(\Pi_{\bullet}^{(+)}\overline{w}),\quad
w\in{\mathcal{W}}.$

Here, for $z\in{\mathcal{Z}}_{\bullet}^{(+)}$, and $a_{\bullet}(z)$ and
$a_{\bullet}^{*}(z)$ are the standard annihilation, resp. creation operators
on $\Gamma_{\mathrm{s}}({\mathcal{Z}}_{\bullet}^{(+)})$, see e.g. [5]. Note
that if $z\in{\mathcal{Z}}_{\bullet}^{(+)}$, then

$\displaystyle\phi_{\bullet}(z)$ $\displaystyle=a_{\bullet}(z),$
$\displaystyle\quad\phi_{\bullet}(\overline{z})$
$\displaystyle=a_{\bullet}^{*}(z);$ $\displaystyle\phi_{\bullet}^{*}(z)$
$\displaystyle=a_{\bullet}^{*}(z),$
$\displaystyle\quad\phi_{\bullet}^{*}(\overline{z})$
$\displaystyle=a_{\bullet}(z).$

###  6.3 Squeezed vectors

Let ${\mathcal{Z}}_{\bullet}$ be a Hilbert space, e.g.,
${\mathcal{Z}}_{\bullet}={\mathcal{Z}}_{\bullet}^{(+)}$ is the particle space
associated to an admissible anti-real involution $S_{\bullet}$. Let $c$ be a
Hilbert–Schmidt operator from $\overline{{\mathcal{Z}}_{\bullet}}$ to
${\mathcal{Z}}_{\bullet}$ satisfying $c^{\\#}=c$ (see (2.8) for the definition
of $c^{\\#}$). Let ${\\{e_{i}\\}}_{i}$ be an orthonormal basis of
${\mathcal{Z}}_{\bullet}$. The operator $c$ can be written as

$\sum_{i,j}\lambda_{ij}|e_{i})(\overline{e_{j}}|,$ (6.2)

where $\mathop{\sum}\limits_{i,j}\lvert\lambda_{ij}\rvert^{2}<\infty$. We then
define

$\displaystyle a_{\bullet}^{*}(c)$
$\displaystyle\coloneqq\sum_{i,j}\lambda_{ij}a_{\bullet}^{*}(e_{i})a_{\bullet}^{*}(e_{j}),$
$\displaystyle a_{\bullet}(c)$
$\displaystyle\coloneqq\sum_{i,j}\overline{\lambda_{ij}}a_{\bullet}(e_{j})a_{\bullet}(e_{i}).$

Clearly, $a_{\bullet}(c)$ and $a_{\bullet}^{*}(c)$ do not depend on the choice
of basis ${\\{e_{i}\\}}_{i}$.

######  Proposition 6.1.

If $\lVert c\rVert<1$, then
$\mathrm{e}^{\frac{1}{2}a_{\bullet}^{*}(c)}\Omega_{\bullet}$ is well-defined.
If $c_{1},c_{2}$ are two such operators, then

$\big{(}\mathrm{e}^{\frac{1}{2}a_{\bullet}^{*}(c_{1})}\Omega_{\bullet}\mkern
2.0mu\big{|}\mkern
2.0mu\mathopen{}\mathrm{e}^{\frac{1}{2}a_{\bullet}^{*}(c_{2})}\Omega_{\bullet}\big{)}=\frac{1}{\sqrt{\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c_{1}^{*}c_{2})}}.$ (6.3)

In particular, the vector

$\Omega_{\bullet,c}\coloneqq\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{\frac{1}{4}}\mathrm{e}^{\frac{1}{2}a_{\bullet}^{*}(c)}\Omega_{\bullet}$
(6.4)

is normalized.

###### Proof.

See e.g. Theorem 11.28 in [5]. ∎

The vector $\Omega_{\bullet,c}$ defined in (6.4) is called a _squeezed
vector_. It satisfies

$\bigl{(}a_{\bullet}(z)-a_{\bullet}^{*}(c\overline{z})\bigr{)}\Omega_{\bullet,c}=0,\quad
z\in{\mathcal{Z}}.$

###  6.4 Implementation of symplectic transformations

Let $R$ be a symplectic (that is, real pseudo-unitary) transformation on
$({\mathcal{W}},Q)$. As before, we fix an anti-real admissible involution
$S_{\bullet}$.

######  Definition 6.2.

We say that an operator $\hat{R}$ _implements $R$_ in the representation
${\mathcal{W}}\ni w\mapsto\phi_{\bullet}(w)$ if it satisfies

$\hat{R}\phi_{\bullet}(w)\hat{R}^{-1}=\phi_{\bullet}(Rw),\quad
w\in{\mathcal{W}}.$

Recall from (4.36), that in the sense of
${\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(+)}}$
we can write $R$ as

$R=\begin{bmatrix}p&q\\\
\overline{p}&\overline{q}\end{bmatrix}=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&d(R)\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}p^{*-1}&0\\\
0&\overline{p}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ c(R)^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$ (6.5)

For brevity we will write $c$, $d$ instead of $c(R)$, $d(R)$.

The following theorem is called the Shale criterion. It is proven e.g. in [5].

######  Theorem 6.3.

The following are equivalent:
(1) $q$ is Hilbert–Schmidt, (2) $pp^{*}-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ is trace class, (3)
$c$ is Hilbert–Schmidt,
(4) $d$ is Hilbert–Schmidt, (5) $R$ is implementable.

If this is the case, then all implementers of $R$ coincide up to a phase
factor. Among them there exists a unique one, called the natural implementer
and denoted $\hat{R}^{\mathrm{nat}}$, which satisfies
$(\Omega|\hat{R}^{\mathrm{nat}}\Omega)>0$. It is equal to

$\hat{R}^{\mathrm{nat}}=(\det
p^{*}p)^{-\frac{1}{4}}\mathrm{e}^{-\frac{1}{2}a_{\bullet}^{*}(d)}\Gamma(p^{*-1})\mathrm{e}^{\frac{1}{2}a_{\bullet}(c)}.$
(6.6)

Unfortunately, the natural implementer defined in (6.6) does not give a
representation of the symplectic group, and only a projective representation.
Under more restrictive conditions one can obtain a $1-2$ representation, by
choosing the metaplectic implementer, see e.g. [5].

######  Proposition 6.4.

If $p-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ is trace class, then the assumptions of
Thm. 6.3 are satisfied. Besides, there exist two _metaplectic implementers_ ,
differing with the sign, which implement $R$:

$\pm\hat{R}^{\mathrm{met}}=\pm(\det
p^{*})^{-\frac{1}{2}}\mathrm{e}^{-\frac{1}{2}a_{\bullet}^{*}(d)}\Gamma(p^{*-1})\mathrm{e}^{\frac{1}{2}a_{\bullet}(c)}.$

###  6.5 Comparison of two Fock representations

Suppose now that $S_{1},S_{2}$ are two admissible anti-real involutions on
${\mathcal{W}}$. Let ${\mathcal{Z}}_{1}^{(+)}$ and ${\mathcal{Z}}_{2}^{(+)}$
be the corresponding particle spaces. Let $\phi_{1}$ and $\phi_{2}$ be the
Fock representations on $\Gamma_{\mathrm{s}}({\mathcal{Z}}_{1}^{(+)})$, resp.
$\Gamma_{\mathrm{s}}({\mathcal{Z}}_{2}^{(+)})$ corresponding to $S_{1}$, resp.
$S_{2}$.

Let $K$, $c$ and $\Upsilon$ be defined as in (4.4) and (4.9) (see also
(4.16)).

######  Proposition 6.5.

  1. (1)

The representations $\phi_{1}$ and $\phi_{2}$ are equivalent if and only if
$c$ is Hilbert–Schmidt.

  2. (2)

Suppose that $c$ is Hilbert–Schmidt. Let
$\Omega_{1}\in\Gamma_{\mathrm{s}}({\mathcal{Z}}_{1}^{(+)})$ be the vacuum in
the $\phi_{1}$ representation. Then the state $\omega_{2}$ coincides with
$(\Omega_{2}\mkern 2.0mu|\mkern 2.0mu\mathopen{}\,\cdot\;\Omega_{2})$, where
$\Omega_{2}$ is the squeezed vector

$\Omega_{2}\coloneqq\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{\frac{1}{4}}\mathrm{e}^{\frac{1}{2}a_{1}^{*}(c)}\Omega_{1}.$
(6.7)

Moreover, we have

$\displaystyle(\Omega_{2}\mkern 2.0mu|\mkern 2.0mu\mathopen{}\Omega_{1})$
$\displaystyle=\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{\frac{1}{4}}=\det\Upsilon^{-\frac{1}{4}},$ (6.8)
$\displaystyle\frac{\big{(}\Omega_{2}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\phi_{2}(v)\phi_{2}^{*}(w)\Omega_{1}\big{)}}{(\Omega_{2}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{1})}$ $\displaystyle=(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}Q\Pi_{2}^{(+)}\Upsilon^{-1}\Pi_{1}^{(+)}w).$
(6.9)

###### Proof.

(1) is proven e.g. in [5].

Let us prove (2). Recall that in (4.12) we defined the operator

$\displaystyle M$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&-c\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{2}}&0\\\ 0&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ -c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}$ (6.10)

satisfying $S_{2}=MS_{1}M^{-1}$. By Thm 6.3 we have

$\displaystyle\hat{M}^{\mathrm{nat}}$ $\displaystyle=\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-\frac{1}{4}}\mathrm{e}^{\frac{1}{2}a_{1}^{*}(c)}\Gamma({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{2}}\mathrm{e}^{-\frac{1}{2}a_{1}(c)},$
(6.11)
$\displaystyle\hat{M}^{\mathrm{nat}}\phi_{1}(v)\hat{M}^{\mathrm{nat}-1}$
$\displaystyle=\phi_{2}(v),\quad\hat{M}^{\mathrm{nat}}\Omega_{1}=\Omega_{2}.$
(6.12)

This implies (6.7) and (6.8).

Let us show (6.9). By (4.8c), $z\in{\mathcal{Z}}_{1}^{(+)}$ implies
$z-\overline{c}z\in{\mathcal{Z}}_{2}^{(+)}$. Therefore, we can write

$\displaystyle\phi_{1}(v)$
$\displaystyle=\phi_{1}\bigl{(}\Pi_{1}^{(+)}v+c\overline{\Pi_{1}^{(+)}}v\bigr{)}+\phi_{1}\bigl{(}\overline{\Pi_{1}^{(+)}}v-c\overline{\Pi_{1}^{(+)}}v\bigr{)}$
$\displaystyle=a_{1}\bigl{(}\Pi_{1}^{(+)}v+c\overline{\Pi_{1}^{(+)}}v\bigr{)}+a_{2}^{*}\bigl{(}\Pi_{1}^{(+)}\overline{v}-\overline{c}\Pi_{1}^{(+)}\overline{v}\bigr{)}$
$\displaystyle\phi_{1}^{*}(w)$
$\displaystyle=\phi_{1}^{*}\bigl{(}\Pi_{1}^{(+)}w\bigr{)}+\phi_{1}^{*}\bigl{(}\overline{\Pi_{1}^{(+)}}w\bigr{)}\quad=\quad
a_{1}^{*}\bigl{(}\Pi_{1}^{(+)}w\bigr{)}+a_{1}\bigl{(}\Pi_{1}^{(+)}\overline{w}\bigr{)}.$

After putting $(\Omega_{2}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\,\cdot\;\Omega_{1})$ around $\phi_{1}(v)\phi_{1}^{*}(w)$, we
move the $a_{2}^{*}$ terms to the left so that they annihilate $\Omega_{2}$,
and the $a_{1}$ terms to the right, so that they annihilate $\Omega_{1}$.
Hence,

$\displaystyle\frac{(\Omega_{2}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\phi_{1}(v)\phi_{1}^{*}(w)\Omega_{1})}{(\Omega_{2}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{1})}$
$\displaystyle=\left[\phi_{1}\bigl{(}\Pi_{1}^{(+)}v+c\overline{\Pi_{1}^{(+)}}v\bigr{)},\phi_{1}^{*}\bigl{(}\Pi_{1}^{(+)}w\bigr{)}\right]$
(6.13)
$\displaystyle=\bigl{(}(\Pi_{1}^{(+)}+c\overline{\Pi_{1}^{(+)}})v|Q\Pi_{1}^{(+)}w\bigr{)}.$
(6.14)

Now, by (4.8a),

$\Pi_{1}^{(+)}+c\overline{\Pi_{1}^{(+)}}=\Pi_{1}^{(+)}\Upsilon^{-1}\Pi_{2}^{(+)}.$

Finally, we use the $Q$-self-adjointness of $\Upsilon$. ∎

######  Remark 6.6.

Let $R$ be a symplectic transformation on $({\mathcal{W}},Q)$. Let $S_{1}$,
$S_{2}$ be two admissible involutions. The matrix representation of $R$ given
in (6.5) can be generalized to the setting
${\mathcal{Z}}_{1}^{(+)}\oplus\overline{{\mathcal{Z}}_{1}^{(+)}}\to{\mathcal{Z}}_{2}^{(+)}\oplus\overline{{\mathcal{Z}}_{2}^{(+)}}$,
as in (4.36).

There exists an obvious generalization of Theorem 6.3 to the context of two
admissible involutions and the corresonding two Fock representations
$\phi_{1}$ and $\phi_{2}$. The only difference is that now implementers
$\hat{R}$ satisfy

$\hat{R}\phi_{1}(w)\hat{R}^{-1}=\phi_{2}(Rw),\quad w\in{\mathcal{W}}.$ (6.15)

and the formula for the natural implementer has now the form

$\hat{R}^{\mathrm{nat}}=(\det
p^{*}p)^{-\frac{1}{4}}\mathrm{e}^{-\frac{1}{2}a_{2}^{*}(d)}\Gamma(p^{*-1})\mathrm{e}^{\frac{1}{2}a_{1}(c)}.$
(6.16)

The physical interpretation of $\hat{R}^{\mathrm{nat}}$ is often the
scattering operator, computed betwee the in Fock representation given by
$S_{1}$ and the out representation give by $S_{2}$. Its matrix elements are
scattering amplitudes, used to compute scattering cross-sections.

If the Shale criterion is satisfied, so that $\hat{R}^{\mathrm{nat}}$ is well
defined, then

$\displaystyle(\det p^{*}p)^{-\frac{1}{4}}=\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+q^{*}q)^{-\frac{1}{4}}=\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{4}}=\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-dd^{*})^{\frac{1}{4}}$ (6.17)

is a positive number less that $1$. If $\hat{R}^{\mathrm{nat}}$ is the
scattering operator, (6.17) has an important physical meaning: it is the
vacuum–vacuum amplitude and equals
$(\Omega_{2}|\hat{R}^{\mathrm{nat}}\Omega_{1})$.

Instead of (6.16), one could introduce the “renormalized Bogoliubov
implementer”

$\hat{R}^{\mathrm{ren}}:=\frac{\hat{R}^{\mathrm{nat}}}{(\Omega|R^{\mathrm{nat}}\Omega)},$
(6.18)

which is always well defined as a quadratic form, even if (6.17) is zero.

If (6.17) is zero, so that $\hat{R}^{\mathrm{nat}}$ is ill-defined, we can
still compute ratios of scattering cross-sections with help of (6.18). Thus a
consequence, in Quantum Field Theory (at least, in its linear version) we do
not need to worry too much about the Shale criterion and the implementability
of the scattering operator.

##  7 Bosonic quantization: complex (or charged) formalism

In this section we will describe the complex or charged formalism of bosonic
quantization At first sight it seems more complicated than the neutral
formalism discussed in the previous section. However there are some points
where it is more convenient than the neutral formalism. For instance,
calculations involving gauge-invariant squeezed vectors of Subsect. 7.3 are
slightly simpler than those using squeezed vectors of Subsect. 6.3.

###  7.1 Charged canonical commutation relations

Suppose that ${\mathcal{W}}$ is a complex vector space equipped with a
Hermitian form

$(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw),\quad v,w\in{\mathcal{W}}.$

Let $\mathrm{CCR}({\mathcal{W}})$ denote the complex unital $*$-algebra
generated by $\psi(w)$ and $\psi^{*}(w)$, $w\in{\mathcal{W}}$, such that

  1. (1)

$\psi^{*}(w)=\psi(w)^{*}$,

  2. (2)

the map ${\mathcal{W}}\ni w\mapsto\psi^{*}(w)$ is linear,

  3. (3)

and the canonical commutation relations hold,

$\bigl{[}\psi(v),\psi^{*}(w)\bigr{]}=(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw),\quad\bigl{[}\psi^{*}(v),\psi^{*}(w)\bigr{]}=0,\quad
v,w\in{\mathcal{W}}.$ (7.1)

###  7.2 Fock representations

Assume, in addition, that $({\mathcal{W}},Q)$ is Krein. Let $S_{\bullet}$ be
an admissible involution on ${\mathcal{W}}$ and introduce
$\Pi_{\bullet}^{(\pm)}$, ${\mathcal{Z}}_{\bullet}^{(\pm)}$ as in Subsect. 2.8.

Then we have a unique centered pure quasi-free state on
$\mathrm{CCR}({\mathcal{W}})$ defined by

$\displaystyle\omega_{\bullet}\bigl{(}\psi(v)\psi^{*}(w)\bigr{)}$
$\displaystyle=(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Q\Pi_{\bullet}^{(+)}w),$
$\displaystyle\omega_{\bullet}\bigl{(}\psi^{*}(v)\psi(w)\bigr{)}$
$\displaystyle=(w\mkern 2.0mu|\mkern 2.0mu\mathopen{}Q\Pi_{\bullet}^{(-)}v),$
$\displaystyle\omega_{\bullet}\bigl{(}\psi^{*}(v)\psi^{*}(w)\bigr{)}$
$\displaystyle=0,$
$\displaystyle\omega_{\bullet}\bigl{(}\psi(v)\psi(w)\bigr{)}$
$\displaystyle=0.$

Let us describe explicitly the GNS representation of $\omega_{\bullet}$. It
acts on the bosonic Fock space

$\Gamma_{\mathrm{s}}\bigl{(}{\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(-)}}\bigr{)}\simeq\Gamma_{\mathrm{s}}({\mathcal{Z}}_{\bullet}^{(+)})\otimes\Gamma_{\mathrm{s}}(\overline{{\mathcal{Z}}_{\bullet}^{(-)}}).$

The state $\omega_{\bullet}$ is represented by the Fock vacuum $(\Omega\mkern
2.0mu|\mkern 2.0mu\mathopen{}\,\cdot\;\Omega)$. Denote the creation and
annihilation operators by $a_{\bullet}^{*}$ and $a_{\bullet}$. The fields
$\psi$ in the GNS representation given by $\omega_{\bullet}$ will be denoted
by $\psi_{\bullet}$. We have

$\displaystyle\psi_{\bullet}(w)$ $\displaystyle\coloneqq
a_{\bullet}(\Pi_{\bullet}^{(+)}w)+a_{\bullet}^{*}(\overline{\Pi_{\bullet}^{(-)}w}),$
$\displaystyle\psi_{\bullet}^{*}(w)$ $\displaystyle\coloneqq
a_{\bullet}^{*}(\Pi_{\bullet}^{(+)}w)+a_{\bullet}(\overline{\Pi_{\bullet}^{(-)}w}).$

The operator $\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})$ plays the
role of a charge:

######  Proposition 7.1.

We have a $U(1)$ group of symmetries

$\displaystyle\mathrm{e}^{\mathrm{i}s\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})}\psi_{\bullet}(w)\mathrm{e}^{-\mathrm{i}s\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})}$
$\displaystyle=\mathrm{e}^{-\mathrm{i}s}\psi_{\bullet}(w),$ (7.2a)
$\displaystyle\mathrm{e}^{\mathrm{i}s\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})}\psi_{\bullet}^{*}(w)\mathrm{e}^{-\mathrm{i}s\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})}$
$\displaystyle=\mathrm{e}^{\mathrm{i}s}\psi_{\bullet}^{*}(w).$ (7.2b)

###### Proof.

We show this for $\psi_{\bullet}$.

$\displaystyle\mathrm{e}^{\mathrm{i}s\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})}\psi_{\bullet}(w)\mathrm{e}^{-\mathrm{i}s\mathop{\mathop{}\\!\mathrm{d}\Gamma}(S_{\bullet})}$
$\displaystyle=a_{\bullet}(\Pi_{\bullet}^{(+)}\mathrm{e}^{\mathrm{i}sS}w)+a_{\bullet}^{*}(\overline{\Pi_{\bullet}^{(-)}\mathrm{e}^{\mathrm{i}sS}w})$
$\displaystyle=a_{\bullet}(\Pi_{\bullet}^{(+)}\mathrm{e}^{\mathrm{i}s}w)+a_{\bullet}^{*}(\overline{\Pi_{\bullet}^{(-)}\mathrm{e}^{\mathrm{i}s}w}),$

where we used that $S$ is an involution. Then the result follows from the
(anti-)linearity of $a_{\bullet}$ and $a_{\bullet}^{*}$. ∎

###  7.3 Gauge invariant squeezed vectors

In typical applications of the charged formalism the evolution and observables
are assumed to be invariant with respect to the $U(1)$ group (7.2). Similarly,
the natural class of squeezed vectors in the charge formalism consists of
gauge invariant squeezed vectors, which we introduce below.

Let $c$ be a Hilbert–Schmidt operator from ${\mathcal{Z}}_{\bullet}^{(-)}$ to
${\mathcal{Z}}_{\bullet}^{(+)}$. Let ${\\{e_{j}\\}}_{i}$ be an orthonormal
basis in ${\mathcal{Z}}_{\bullet}^{(-)}$ and ${\\{f_{i}\\}}_{i}$ be an
orthonormal basis in ${\mathcal{Z}}_{\bullet}^{(+)}$. The operator $c$ can be
written as

$\sum_{i,j}c_{ij}|f_{i})(e_{j}|,$ (7.3)

where $\sum\limits_{i,j}|c_{ij}|^{2}<\infty$. We then define

$\displaystyle a_{\bullet,\mathrm{gi}}^{*}(c)$
$\displaystyle\coloneqq\sum_{i,j}c_{ij}a_{\bullet}^{*}(f_{i})a_{\bullet}^{*}(\overline{e_{j}}),$
$\displaystyle a_{\bullet,\mathrm{gi}}(c)$
$\displaystyle\coloneqq\sum_{i,j}\overline{c}_{ij}a_{\bullet}(\overline{e_{j}})a_{\bullet}(f_{i}).$

Clearly, neither $a_{\bullet,\mathrm{gi}}^{*}(c)$ nor
$a_{\bullet,\mathrm{gi}}(c)$ depend on the bases ${\\{e_{i}\\}}_{i}$ and
${\\{f_{i}\\}}_{i}$.

######  Proposition 7.2.

If $\lVert c\rVert<1$, then
$\mathrm{e}^{a_{\bullet,\mathrm{gi}}^{*}(c)}\Omega_{\bullet}$ is well-defined
as a vector in
$\Gamma_{\mathrm{s}}\bigl{(}{\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(-)}}\bigr{)}$.
Moreover, if $c_{1}$, $c_{2}$ are two such operators, then

$\big{(}\mathrm{e}^{a_{\bullet,\mathrm{gi}}^{*}(c_{1})}\Omega_{\bullet}\mkern
2.0mu\big{|}\mkern
2.0mu\mathopen{}\mathrm{e}^{a_{\bullet,\mathrm{gi}}^{*}(c_{2})}\Omega_{\bullet}\big{)}=\frac{1}{\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c_{1}^{*}c_{2})}.$ (7.4)

In particular, the vector

$\Omega_{\bullet,\mathrm{gi}}(c)\coloneqq\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{\frac{1}{2}}\mathrm{e}^{a_{\bullet,\mathrm{gi}}^{*}(c)}\Omega_{\bullet}$
(7.5)

is normalized.

(7.5) is called a _gauge-invariant squeezed vector_. It satisfies

$\displaystyle\bigl{(}a_{\bullet,\mathrm{gi}}(\overline{w})-a_{\bullet,\mathrm{gi}}^{*}(cw)\bigr{)}\Omega_{\bullet,\mathrm{gi}}(c)$
$\displaystyle=0,\quad w\in{\mathcal{Z}}_{\bullet}^{(-)},$
$\displaystyle\bigl{(}a_{\bullet,\mathrm{gi}}(w)-a_{\bullet,\mathrm{gi}}^{*}(\overline{c^{*}w})\bigr{)}\Omega_{\bullet,\mathrm{gi}}(c)$
$\displaystyle=0,\quad w\in{\mathcal{Z}}_{\bullet}^{(+)}.$

###  7.4 Comparison of squeezed vectors in the real and complex formalism

Gauge-invariant squeezed vectors can be treated as usual ones, introduced in
Subsect. 6.3. Recall that to define a squeezed vector in the charged formalism
we consider the Fock space
$\Gamma_{\mathrm{s}}({\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(-)}})$
and a Hilbert–Schmidt operator
$c:{\mathcal{Z}}_{\bullet}^{(-)}\to{\mathcal{Z}}_{\bullet}^{(+)}$. We set
${\mathcal{Z}}_{\bullet}\coloneqq{\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(-)}}$
and consider

$\tilde{c}\coloneqq\begin{bmatrix}0&c\\\ c^{\\#}&0\end{bmatrix},$

which is an operator
$\overline{\mathcal{Z}}_{\bullet}\to{\mathcal{Z}}_{\bullet}$ such that
${\tilde{c}}^{\\#}=\tilde{c}$. Now,

$\displaystyle a_{\bullet,\mathrm{gi}}^{*}(c)$
$\displaystyle=\frac{1}{2}a_{\bullet}^{*}(\tilde{c}),$ (7.6)
$\displaystyle\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-c^{*}c)^{2}$
$\displaystyle=\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-{\tilde{c}}^{*}\tilde{c}),$ (7.7)
$\displaystyle\Omega_{\bullet,\mathrm{gi}}(c)$
$\displaystyle=\Omega_{\bullet,\tilde{c}}.$ (7.8)

Therefore, the formulas (7.4) and (6.3) are consistent with one another.

###  7.5 Implementation of pseudo-unitary transformations

Let $R$ be a pseudo-unitary transformation on $({\mathcal{W}},Q)$. As before,
we fix an admissible involution $S_{\bullet}$.

######  Definition 7.3.

We say that an operator $\hat{R}$ _implements $R$_ in the representation
${\mathcal{W}}\ni w\mapsto\psi_{\bullet}^{*}(w)$ if it satisfies

$\hat{R}\psi_{\bullet}^{*}(w)\hat{R}^{-1}=\phi_{\bullet}^{*}(Rw),\quad
w\in{\mathcal{W}}.$

Recall from (4.27), that in the sense of
${\mathcal{Z}}_{\bullet}^{(+)}\oplus{\mathcal{Z}}_{\bullet}^{(-)}$ we can
write $R$ as

$\displaystyle R$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&d\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}R_{++}^{*-1}&0\\\
0&R_{--}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}$ (7.9)
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
d^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}R_{++}&0\\\
0&R_{--}^{*-1}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&c\\\ &{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(7.10)

For brevity we will write $c$, $d$ instead of $c(R)$, $d(R)$.

The following theorem is a complex version of Thm 6.3.

######  Theorem 7.4.

The following are equivalent:
(1) $R_{+-}$ is Hilbert–Schmidt, (2) $R_{-+}$ is Hilbert-Schmidt,
(3) $R_{++}R_{++}^{*}-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ is trace class, (4)
$R_{--}R_{--}^{*}-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ is trace class,
(5) $c$ is Hilbert–Schmidt, (6) $d$ is Hilbert–Schmidt, (7) $R$ is
implementable.
If this is the case, then all implementers of $R$ coincide up to a phase
factor. Among them there exists a unique one, called the natural implementer
and denoted $\hat{R}^{\mathrm{nat}}$, which satisfies
$(\Omega|\hat{R}^{\mathrm{nat}}\Omega)>0$. It is equal to

$\hat{R}^{\mathrm{nat}}=\lvert\det
R_{++}^{*}R_{--}^{\\#}\rvert^{-\frac{1}{2}}\mathrm{e}^{-a_{\bullet\mathrm{gi}}^{*}(d)}\Gamma(R_{++}^{*-1}\oplus
R_{--}^{\\#-1})\mathrm{e}^{a_{\bullet\mathrm{gi}}(c)}.$

###### Proof.

Take the complex conjugate of (7.10) and reverse the order of the components,
obtaining, in the sense of
$\overline{{\mathcal{Z}}_{\bullet}^{(-)}}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(+)}}$,

$\displaystyle R$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&d^{\\#}\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}R_{--}^{\\#-1}&0\\\
0&\overline{R}_{++}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ \overline{c}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(7.11)

Then insert (7.11) in the middle of (7.9), obtaining the operator on
${\mathcal{Z}}_{\bullet}\oplus\overline{{\mathcal{Z}}_{\bullet}}$, where
${\mathcal{Z}}_{\bullet}\coloneqq{\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(-)}}$:

$\displaystyle R$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0&0&d\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&d^{\\#}&0\\\
0&0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\ 0&0&0&{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}R_{++}^{*-1}&0&0&0\\\
0&R_{--}^{\\#-1}&0&0\\\ 0&0&\overline{R}_{++}&0\\\
0&0&0&R_{--}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0&0&0\\\
0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0&0\\\ 0&\overline{c}&{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ c^{*}&0&0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}$ (7.12)

Then we apply Thm. 6.3 from the neutral formalism, and take into account
(7.6). ∎

Here is the complex version of Prop. 6.4:

######  Proposition 7.5.

If $R_{++}-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ and $R_{--}-{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$
are trace class, the assumptions of Thm. 6.3 are satisfied. Besides, there
exist two _metaplectic implementers_ , differing with the sign:

$\pm\hat{R}^{\mathrm{met}}=\bigl{(}\det
R_{++}^{*}R_{--}^{\\#}\bigr{)}^{-\frac{1}{2}}\mathrm{e}^{-a_{\bullet,\mathrm{gi}}^{*}(d)}\Gamma(R_{++}^{*-1}\oplus
R_{--}^{\\#-1})\mathrm{e}^{a_{\bullet,\mathrm{gi}}(c)}.$

###  7.6 Comparison of two Fock representations

Suppose now that $S_{1},S_{2}$ are two admissible involutions on
${\mathcal{W}}$. Let ${\mathcal{Z}}_{1}^{(\pm)}$ and
${\mathcal{Z}}_{2}^{(\pm)}$ be the corresponding particle spaces,
$\psi_{1}^{*}$ and $\phi_{2}^{*}$ be the Fock representations, etc. Let $K$,
$c$ and $\Upsilon$ be defined as in (4.4) and (4.9).

The following proposition is the complex version of Prop. 6.5. We state it in
detail and sketch the proof, since there are some subtle differences between
the real and complex case.

######  Proposition 7.6.

  1. (1)

The representations $\psi_{1}^{*}$ and $\psi_{2}^{*}$ are equivalent if and
only if $c$ is Hilbert–Schmidt.

  2. (2)

Suppose that $c$ is Hilbert–Schmidt. Let
$\Omega_{1}\in\Gamma_{\mathrm{s}}({\mathcal{Z}}_{1}^{(+)}\oplus\overline{{\mathcal{Z}}_{1}^{(-)}})$
be the vacuum in the $\psi_{1}^{*}$ representation. Then the state
$\omega_{2}$ coincides with $(\Omega_{2}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\,\cdot\;\Omega_{2})$, where $\Omega_{2}$ is the squeezed
vector

$\Omega_{2}\coloneqq\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{\frac{1}{2}}\mathrm{e}^{a_{\mathrm{gi}1}^{*}(c)}\Omega_{1}.$
(7.13)

Moreover, we have

$\displaystyle(\Omega_{2}\mkern 2.0mu|\mkern 2.0mu\mathopen{}\Omega_{1})$
$\displaystyle=\det({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{\frac{1}{2}}=\det\Upsilon^{-\frac{1}{2}},$ (7.14)
$\displaystyle\frac{\big{(}\Omega_{2}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\psi_{2}(v)\psi_{2}^{*}(w)\Omega_{1}\big{)}}{(\Omega_{2}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{1})}$ $\displaystyle=(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}Q\Pi_{2}^{(+)}\Upsilon^{-1}\Pi_{1}^{(+)}w).$
(7.15)

###### Proof.

Let us prove (2). Note that the operator $M$ defined in (4.12) and recalled in
(6.10) can still be used. It satisfies satisfying $S_{2}=MS_{1}M^{-1}$. By Thm
7.4 we have

$\displaystyle\hat{M}^{\mathrm{nat}}$ $\displaystyle=\det({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-\frac{1}{2}}\mathrm{e}^{a_{\mathrm{gi}1}^{*}(c)}\Gamma\big{(}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{2}}\oplus({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{2}}\big{)}\mathrm{e}^{-a_{\mathrm{gi}1}(c)},$
(7.16)
$\displaystyle\hat{M}^{\mathrm{nat}}\psi_{1}^{*}(v)\hat{M}^{\mathrm{nat}-1}$
$\displaystyle=\psi_{2}^{*}(v),\quad\hat{M}^{\mathrm{nat}}\Omega_{1}=\Omega_{2}.$
(7.17)

This implies (7.13) and (7.14).

Let us show (6.9). By (4.8b), $z\in{\mathcal{Z}}_{1}^{(-)}$ implies
$z-cz\in{\mathcal{Z}}_{2}^{(-)}$. Therefore, we can write

$\displaystyle\psi_{1}(v)$
$\displaystyle=\psi_{1}\bigl{(}\Pi_{1}^{(+)}v+c\Pi_{1}^{(-)}v\bigr{)}+\psi_{1}\bigl{(}\Pi_{1}^{(-)}v-c\Pi_{1}^{(-)}v\bigr{)}$
$\displaystyle=a_{1}\bigl{(}\Pi_{1}^{(+)}v+c\Pi_{1}^{(-)}v\bigr{)}+a_{2}^{*}\bigl{(}\overline{\Pi_{1}^{(+)}v-c\Pi_{1}^{(+)}v}\bigr{)}$
$\displaystyle\psi_{1}^{*}(w)$
$\displaystyle=\psi_{1}^{*}\bigl{(}\Pi_{1}^{(+)}w\bigr{)}+\psi_{1}^{*}\bigl{(}\Pi_{1}^{(-)}w\bigr{)}\quad=\quad
a_{1}^{*}\bigl{(}\Pi_{1}^{(+)}w\bigr{)}+a_{1}\bigl{(}\overline{\Pi_{1}^{(+)}w}\bigr{)}.$

After putting $(\Omega_{2}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\,\cdot\;\Omega_{1})$ around $\psi_{1}(v)\psi_{1}^{*}(w)$, we
move the $a_{2}^{*}$ terms to the left so that they annihilates $\Omega_{2}$,
and the $a_{1}$ terms to the right, so that they annihilate $\Omega_{1}$.
Hence,

$\displaystyle\frac{(\Omega_{2}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\psi_{1}(v)\psi_{1}^{*}(w)\Omega_{1})}{(\Omega_{2}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{1})}$
$\displaystyle=\left[\psi_{1}\bigl{(}\Pi_{1}^{(+)}v+c\Pi_{1}^{(-)}v\bigr{)},\psi_{1}^{*}\bigl{(}\Pi_{1}^{(+)}w\bigr{)}\right]$
(7.18)
$\displaystyle=\bigl{(}(\Pi_{1}^{(+)}+c\Pi_{1}^{(+)})v|Q\Pi_{1}^{(+)}w\bigr{)}.$
(7.19)

Finally, by (4.8a),

$\Pi_{1}^{(+)}+c\Pi_{1}^{(-)}=\Pi_{1}^{(+)}\Upsilon^{-1}\Pi_{2}^{(+)}.\qed$

Remark 6.6 are still valid in the complex setting with obvious changes.

##  8 Klein-Gordon equation and Quantum Field Theory
on curved space-times

In this section we formulate the main results of this paper in the setting of
QFT on curved spacetimes, more precisely, on a globally hyperbolic manifolds
equipped with electromagnetic and scalar potentials. We describe the role
various propagators play in the theory of quantum fields satisfying the Klein-
Gordon equation.

The usual presentations of this topic make the assumption that all
coefficients in the Klein-Gordon equation are smooth. The results that we have
obtained in the previous sections allow us to consider systems with much lower
regularity. However, it is not very convenient to do so. Besides, in practice
the corresponding conditions need to be formulated in a non-covariant way,
which may seem not very elegant. Therefore, in this section we assume that
everything is smooth. We do not specify all technical conditions, which the
reader can easily determine, if needed, based on the theory described in
previous sections.

Most of the time, $M$ is a globally hyperbolic manifold, see e.g. [1]. If
$\Omega\subset M$, then $J^{\vee}(\Omega)$ denotes the future shadow, and
$J^{\wedge}(\Omega)$ the past shadow of $\Omega$, that is, the set of all
points in $M$ that can be reached from $\Omega$ by future/past directed causal
paths. A set $\Theta\subset M$ is called space compact if there exists a
compact $\Omega\subset M$ such that $\Theta\subset J^{\vee}(\Omega)\cup
J^{\wedge}(\Omega)$. $C_{\mathrm{sc}}(M)$ denotes the set of continuous
functions on $M$ with a space compact support.

###  8.1 Half-densities on a pseudo-Riemannian manifold

Let $M$ be a manifold. A half-density is an assignment of a complex function
to every coordinate patch satisfying the following condition: if $x\mapsto
f(x)$ and $x^{\prime}\mapsto f^{\prime}(x^{\prime})$ are two such assignments,
then

$\Big{|}\frac{\partial x^{\prime}}{\partial
x}\Big{|}^{\frac{1}{2}}f^{\prime}(x^{\prime})=f(x),$ (8.1)

where $\Big{|}\frac{\partial x^{\prime}}{\partial x}\Big{|}$ denotes the
Jacobian of the change of coordinates. The space of square integrable half-
densitied will be denoted $L^{2}(M)$. Thus if we choose coordinates $x$ and
the support of a function $f$ is contained in the corresponding coordinate
patch, then

$(f_{1}|f_{2})=\int\overline{f_{1}(x)}f_{2}(x)\mathop{}\\!\mathrm{d}x,$

where
$\mathop{}\\!\mathrm{d}x=\mathop{}\\!\mathrm{d}x^{1}\cdots\mathop{}\\!\mathrm{d}x^{d}$
is the Lebesgue measure.

Suppose that $M$ is a pseudo-Riemannian manifold $M$ with a metric tensor,
which in coordinates $x$ is given by the matrix $[g_{\mu\nu}(x)]$. Let
$|g|(x):=\big{|}\det[g_{\mu\nu}(x)]\big{|}$. $M$ has a distinguished density,
which in the coordinates $x$ is given by
$\sqrt{|g|(x)}\mathop{}\\!\mathrm{d}x$. The space of scalar functions, square
integrable wrt $\sqrt{|g|(x)}\mathop{}\\!\mathrm{d}x$, is denoted
$L^{2}(M,\sqrt{|g|})$. Obviously

$L^{2}(M,\sqrt{|g|})\ni f\mapsto|g|^{\frac{1}{4}}f\in L^{2}(M)$

is a unitary map.

###  8.2 Klein-Gordon equation on spacetime and the conserved current

Suppose a pseudo-Riemannian manifold is equipped with a vector field
$[A^{\mu}(x)]$, called the electromagnetic potential, and the scalar potential
$Y(x)$.

The Klein-Gordon operator, written first in the scalar and then in the half-
density formalism is

$\displaystyle K$ $\displaystyle\coloneqq-\lvert
g\rvert^{-\frac{1}{2}}(D_{\mu}-A_{\mu})\lvert
g\rvert^{\frac{1}{2}}g^{\mu\nu}(D_{\nu}-A_{\nu})-Y,$ (8.2) $\displaystyle K$
$\displaystyle\coloneqq-\lvert g\rvert^{-\frac{1}{4}}(D_{\mu}-A_{\mu})\lvert
g\rvert^{\frac{1}{2}}g^{\mu\nu}(D_{\nu}-A_{\nu})\lvert
g\rvert^{-\frac{1}{4}}-Y.$ (8.3)

For functions $u,v\in C^{\infty}(M)$ introduce the current, again first in the
scalar, then in the half-density formalism:

$\displaystyle j^{\mu}(x;\overline{u},v):=$
$\displaystyle-\overline{u(x)}g^{\mu\nu}(x)|g|^{\frac{1}{2}}(x)\big{(}D_{\nu}-A_{\nu}(x)\big{)}v(x)$
$\displaystyle-\overline{\big{(}D_{\nu}-A_{\nu}(x)\big{)}u(x)}g^{\mu\nu}(x)|g|^{\frac{1}{2}}(x)v(x),$
(8.4) $\displaystyle j^{\mu}(x;\overline{u},v):=$
$\displaystyle-\overline{u(x)}g^{\mu\nu}(x)|g|^{\frac{1}{4}}(x)(D_{\nu}-A_{\nu}(x))|g|^{-\frac{1}{4}}(x)v(x)$
$\displaystyle-\overline{\big{(}D_{\nu}-A_{\nu}(x)\big{)}|g|^{-\frac{1}{4}}(x)u(x)}g^{\mu\nu}(x)|g|^{\frac{1}{4}}(x)v(x).$
(8.5)

We check that if $u$, $v$ solve the Klein-Gordon equation, that is

$Ku=Kv=0,$ (8.6)

then the current $j^{\mu}(x,u,v)$ is conserved, that is

$\partial_{\mu}j^{\mu}(x,u,v)=0.$ (8.7)

Let $M$ be globally hyperbolic. Let $\mathrm{Sol}_{\mathrm{sc}}^{\infty}$
denote the space of smooth space compact solutions of the Klein-Gordon
equation. Let $u,v\in\mathrm{Sol}_{\mathrm{sc}}^{\infty}$. Then

$(u|Qv):=\int_{\mathcal{S}}j^{\mu}(x,\overline{u},v)\mathop{}\\!\mathrm{d}\mathrm{s}_{\mu}(x)$
(8.8)

does not depend on the choice of a Cauchy surface $\mathcal{S}$, where
$\mathop{}\\!\mathrm{d}\mathrm{s}_{\mu}(x)$ denotes the natural measure on
$\mathcal{S}$ times the normal vector. (8.8) defines a symplectic form on
$\mathrm{Sol}_{\mathrm{sc}}^{\infty}$.

###  8.3 Foliating the spacetime

We assume that $M$ is equipped with a diffeomorphism $I\times\Sigma\to M$,
where $I$ is, as usual, $]t_{-},t_{+}[$, and $\Sigma$ is a manifold. In other
words, $M$ is equipped with a _time function_ $t\in I$, and all the leaves of
the foliation $\Sigma_{t}=\\{t\\}\times\Sigma$ are identified with a fixed
manifold $\Sigma$.

The restriction of $g$ to the tangent space of $\Sigma_{t}$ defines a time-
dependent family of metrics on $\Sigma$, denoted $h(t)$. We make the
assumption that all $h(t)$ are Riemannian, or, what is equivalent, that the
covector $\mathop{}\\!\mathrm{d}t$ is always timelike. In coordinates, the
metric and the inverse metric can be written as

$\displaystyle
g_{\mu\nu}\mathop{}\\!\mathrm{d}x^{\mu}\mathop{}\\!\mathrm{d}x^{\nu}$
$\displaystyle=-\alpha^{2}\mathop{}\\!\mathrm{d}t^{2}+h_{ij}(\mathop{}\\!\mathrm{d}x^{i}+\beta^{i}\mathop{}\\!\mathrm{d}t)(\mathop{}\\!\mathrm{d}x^{j}+\beta^{j}\mathop{}\\!\mathrm{d}t),$
$\displaystyle g^{\mu\nu}\partial_{\mu}\partial_{\nu}$
$\displaystyle=-\frac{1}{\alpha^{2}}(\partial_{t}-\beta^{i}\partial_{i})^{2}+h^{ij}\partial_{i}\partial_{j}.$

The generic notation for a point of $M$ will be $(t,\vec{x})$. Note that

$L^{2}(M)=L^{2}\big{(}I,{\mathcal{K}}\big{)}\simeq
L^{2}(I)\otimes{\mathcal{K}},$ (8.9)

where ${\mathcal{K}}=L^{2}(\Sigma)$ is the space of square integrable half-
densities on $\Sigma$.

The Klein-Gordon operator can now be written

$\displaystyle K$ $\displaystyle=\lvert
g\rvert^{-\frac{1}{4}}(D_{0}-\beta^{i}D_{i}-A_{0}+\beta^{i}A_{i})\frac{\lvert
g\rvert^{\frac{1}{2}}}{\alpha^{2}}(D_{0}-\beta^{i}D_{i}-A_{0}+\beta^{i}A_{i})\lvert
g\rvert^{-\frac{1}{4}}$ $\displaystyle-\lvert
g\rvert^{-\frac{1}{4}}(D_{i}-A_{i})\lvert
g\rvert^{\frac{1}{2}}h^{ij}(D_{j}-A_{j})\lvert g\rvert^{-\frac{1}{4}}-Y$
$\displaystyle=(D_{0}+W^{*})\frac{1}{\alpha^{2}}(D_{0}+W)-L,$ (8.10)

where

$\displaystyle L$ $\displaystyle:=\lvert
g\rvert^{-\frac{1}{4}}(D_{i}-A_{i})\lvert
g\rvert^{\frac{1}{2}}h^{ij}(D_{j}-A_{j})\lvert g\rvert^{-\frac{1}{4}}+Y$
(8.11) $\displaystyle W$
$\displaystyle:=\beta^{i}D_{i}-A_{0}+\beta^{i}A_{i}+\frac{\mathrm{i}}{4}\lvert
g\rvert^{-1}\lvert g\rvert_{,0}-\frac{\mathrm{i}}{4}\beta^{i}\lvert
g\rvert^{-1}\lvert g\rvert_{,i}$ (8.12)

The temporal component of the current is

$\displaystyle j^{0}(x,\overline{u},v)=$
$\displaystyle\overline{u(x)}\frac{1}{\alpha^{2}(x)}\big{(}D_{0}-W(x)\big{)}v(x)$
$\displaystyle+\overline{\frac{1}{\alpha^{2}(x)}\big{(}D_{0}-W(x)\big{)}u(x)}v(x).$
(8.13)

Therefore, for any $t\in I$,

$\displaystyle(u|Qv)$
$\displaystyle=\int_{\Sigma}\overline{u(t,\vec{x})}\frac{1}{\alpha^{2}(t,\vec{x})}(D_{0}+W\big{(}t,\vec{x})\big{)}v(t,\vec{x})\mathop{}\\!\mathrm{d}\vec{x}$
$\displaystyle+\int_{\Sigma}\overline{(D_{0}+W\big{(}t,\vec{x})\big{)}u(t,\vec{x})}\frac{1}{\alpha^{2}(t,\vec{x})}v(t,\vec{x})\mathop{}\\!\mathrm{d}\vec{x}.$
(8.14)

###  8.4 Space of solution to the Klein-Gordon equation

After identifying $M\simeq I\times\Sigma$, (8.9) and (8.10) show that we are
in the setting of Sect. 5 about the abstract Klein Gordon operator. Let us
impose Assumption 5.1. We can introduce all the formalism of Sect. 5, such as
the spaces ${\mathcal{K}}_{\beta}$ and ${\mathcal{W}}_{\lambda}$, the
generator of the evolution $B(t)$ and the evolution itself $R(t,s)$.

Let us define the space $\mathrm{Sol}$ to be the space of functions $M\simeq
I\times\Sigma\ni(t,\vec{x})\mapsto u(t,\vec{x})\in\mathbb{C}$ such that $I\ni
t\mapsto u(t,\cdot)\in C(I,{\mathcal{K}}_{\frac{1}{2}})$ and for some (hence
all) $t\in I$ there exists $w_{t}\in{\mathcal{W}}_{0}$ such that
$u(s,\cdot)=\big{(}R(s,t)w_{t}\big{)}_{1}$. Clearly, $w_{t}$ is uniquely
defined, and we will write $\gamma_{t}(u)$ for $w_{t}$. At least formally, we
have,

$\gamma_{t}(u)(\vec{x}):=\begin{bmatrix}\gamma_{t}(u)_{1}(\vec{x})\\\
\gamma_{t}(u)_{2}(\vec{x})\end{bmatrix}=\begin{bmatrix}u(t,\vec{x})\\\
-\frac{1}{\alpha^{2}(t,\vec{x})}(D_{0}-W(t,\vec{x}))u(t,\vec{x})\end{bmatrix}.$
(8.15)

Clearly,

$R(t,s)\gamma_{s}(u)=\gamma_{t}(u)$ (8.16)

Thus $\mathrm{Sol}$ is the space of space-time solutions of the Klein-Gordon
equation obtained by the formalism of Sect. 5. It is naturally a Krein space
with the Krein structure transported from ${\mathcal{W}}_{0}$ by any of the
isomorphisms $\gamma_{t}$:

$(u|Qv)=\big{(}\gamma_{t}(u)|Q\gamma_{t}(v)\big{)},\quad u,v\in\mathrm{Sol},$
(8.17)

where (8.17) does not depend on the choice of $t$.

The space $\mathrm{Sol}_{\mathrm{sc}}^{\infty}$ is clearly dense in
$\mathrm{Sol}$. Besides, (8.14) coincides with (8.17) on
$\mathrm{Sol}_{\mathrm{sc}}^{\infty}$.

###  8.5 Classical propagators

The existence of classical propagators is a classic subject covered in many
sources. It holds in a rather great generality. Let us recall the main facts
about classical propagators.

######  Theorem 8.1.

Let $M$ be a (smooth) globally hyperbolic manifold, and $[A^{\mu}]$, $Y$
smooth functions. Then for any $f\in C_{\mathrm{c}}^{\infty}(M)$ there exist
unique $u^{\vee},u^{\wedge}\in C_{\mathrm{sc}}^{\infty}(M)$ such that

$\displaystyle Ku^{\vee}$ $\displaystyle=Ku^{\wedge}=f,$ (8.18)
$\displaystyle\operatorname{supp}u^{\vee}\cap
J^{\wedge}\big{(}\operatorname{supp}(f)\big{)}=\emptyset,$
$\displaystyle\quad\operatorname{supp}u^{\wedge}\cap
J^{\vee}\big{(}\operatorname{supp}(f)\big{)}=\emptyset.$ (8.19)

These solutions satisfy

$\displaystyle\operatorname{supp}u^{\vee}\subset
J^{\vee}\big{(}\operatorname{supp}(f)\big{)},\quad\operatorname{supp}u^{\wedge}\subset
J^{\wedge}\big{(}\operatorname{supp}(f)\big{)}.$ (8.20)

Moreover, there exist unique distributions $G^{\vee}$ and $G^{\wedge}$ on
$M\times M$ such that

$\displaystyle u^{\vee}(x)=\int G^{\vee}(x,y)f(y)\mathop{}\\!\mathrm{d}y,\quad
u^{\wedge}(x)=\int G^{\wedge}(x,y)f(y)\mathop{}\\!\mathrm{d}y.$ (8.21)

Let us remark that the method of evolution equations can be used in a prooof
of the above theorem. In fact, let us introduce the identification $M\simeq
I\times\Sigma$ and impose Assumption 5.1. Then we can define the classical
propagators $G^{\bullet}$, $\bullet=\vee,\wedge,\mathrm{PJ}$ as operators
$C_{\mathrm{c}}^{\infty}(I,\Sigma))\to C_{\mathrm{c}}^{\infty}(I,\Sigma))$. By
the Schwartz kernel theorem they possess distributional kernels, which we
denote $G^{\bullet}(x,y)=G^{\bullet}(t,\vec{x},s,\vec{y})$, Clearly the
support of

$u^{\bullet}(t,\vec{x}):=\int
G^{\bullet}(t,\vec{x};s,\vec{y})f(s,\vec{y})\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}\vec{y}$
(8.22)

is contained in $t$ later than $\operatorname{supp}(f)$ for $\bullet=\vee$ and
in $t$ earlier than $\operatorname{supp}(f)$ for $\bullet=\wedge$. Hence
(8.19) holds. Besides, (8.18) is true. By the uniqueness, (8.22) coincides
with $u^{\vee}$, resp. $u^{\wedge}$.

The following representation of elements of $\mathrm{Sol}$ is useful: Let
$f\in C_{\mathrm{c}}^{\infty}(M)$. Define $G^{\mathrm{PJ}}f$ by

$G^{\mathrm{PJ}}f(x)=\int G^{\mathrm{PJ}}(x,y)f(y)\mathop{}\\!\mathrm{d}y.$
(8.23)

######  Proposition 8.2.

$G^{\mathrm{PJ}}f$ belongs to $\mathrm{Sol}$ and

$\frac{1}{2}\big{(}G^{\mathrm{PJ}}f_{1}|QG^{\mathrm{PJ}}f_{2}\big{)}=\iint\overline{f(x_{1})}G^{\mathrm{PJ}}(x_{1},x_{2})f(x_{2})\mathop{}\\!\mathrm{d}x_{1}\mathop{}\\!\mathrm{d}x_{2}.$
(8.24)

###  8.6 Non-classical propagators

We will use the identification $M\simeq I\times\Sigma$ and impose Assumption
5.1. If $I$ is finite, we choose two admissible involutions $S_{\pm}$ on
${\mathcal{W}}_{0}$.

If $I=\mathbb{R}$, we assume that the spacetime is stationary for large times
and assume that $B(\pm\infty)$ are stable. We set
$S_{\pm}:=\operatorname{sgn}\big{(}B(\pm\infty)\big{)}$, which are
automatically admissible involutions.

We define $\Pi_{\pm}^{(+)},\Pi_{\pm}^{(-)}$, as well as
$\Pi_{\pm}^{(+)}(t),\Pi_{\pm}^{(-)}(t)$ in the usual way. We also define the
non-classical propagators $G_{\pm}^{(+)},$ $G_{\pm}^{(+)},$ $G^{(+-)},$
$G^{(-+)}$.

If $I$ is finite, the non-classical propagators can be interpreted as bounded
operators on $L^{2}(M)$. If $G^{(+-)},$ $G^{(-+)}$ have zero nullspaces, then
we can define

$K^{(+-)}:=G^{(+-)-1},\quad K^{(-+)}:=G^{(-+)-1},$ (8.25)

which can be treated as well-posed realizations of the Klein-Gordon operator
satisfying $K^{(+-)*}=K^{(-+)*}$.

If $I=\mathbb{R}$, then the non-classical propagators can be understood as,
say, operators $C_{\mathrm{c}}(I,L^{2}(\Sigma))\to C(I,L^{2}(\Sigma))$.

It is clear that the operator $K$ is Hermitian (or, as it is often termed,
symmetric) on $C_{\mathrm{c}}^{\infty}(M)$. One can ask about the existence of
its self-adjoint extensions. This is the subject of following conjecture,
which is essentially a spacetime version of Conjecture 8.3.

######  Conjecture 8.3.

For a large class of asymptotically stationary Klein-Gordon operators the
following holds:

  1. (1)

The operator $K$ with the domain $C_{\mathrm{c}}^{\infty}(M)$ is essentially
self-adjoint in the sense of $L^{2}(M)$. Denote its unique self-adjoint
extension by $K^{\mathrm{s.a.}}$.

  2. (2)

In the sense of $\langle t\rangle^{-s}L^{2}(M)\to\langle
t\rangle^{s}L^{2}(M)$, for $s>\frac{1}{2}$,

$\displaystyle\operatorname*{s-lim}_{\epsilon\searrow
0}(K^{\mathrm{s.a.}}-\mathrm{i}\epsilon)^{-1}$ $\displaystyle=G^{{(+-)}},$
(8.26) $\displaystyle\operatorname*{s-lim}_{\epsilon\searrow
0}(K^{\mathrm{s.a.}}+\mathrm{i}\epsilon)^{-1}$ $\displaystyle=G^{{(-+)}}.$
(8.27)

Note that Conjecture 8.3 is true in the stationary case, see [6] and [7]. As
proven by A. Vasy, it is also true if the space-time is asymptotically
Minkowskian, see [25].

###  8.7 Charged quantum fields

Let us apply the formalism of charged quantization, described in Subsection
7.1, to the Krein space $\mathrm{Sol}$. Thus we obtain a linear map

$\mathrm{Sol}\ni v\mapsto\psi^{*}(v)=\psi(v)^{*},$ (8.28)

satisfying

$\bigl{[}\psi(v),\psi^{*}(w)\bigr{]}=(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw),\quad\bigl{[}\psi^{*}(v),\psi^{*}(w)\bigr{]}=0,\quad
v,w\in{\mathcal{W}}.$ (8.29)

$\psi^{*}(v)$ and $\psi(v)$ are called smeared-out charged quantum fields.

We fix all the data of Subsect. 8.6. We define the Fock states $\omega_{\pm}$
by specifying:

$\displaystyle\omega_{\pm}\bigl{(}\psi(v)\psi^{*}(w)\bigr{)}$
$\displaystyle=(\gamma_{t}(v)\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q\Pi_{\pm}^{(+)}(t)\gamma_{t}(w)),$
$\displaystyle\omega_{\pm}\bigl{(}\psi^{*}(v)\psi(w)\bigr{)}$
$\displaystyle=(\gamma_{t}(w)\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q\Pi_{\pm}^{(-)}(t)\gamma_{t}(v)),$
$\displaystyle\omega_{\pm}\bigl{(}\psi^{*}(v)\psi^{*}(w)\bigr{)}$
$\displaystyle=0,$ $\displaystyle\omega_{\pm}\bigl{(}\psi(v)\psi(w)\bigr{)}$
$\displaystyle=0,$

where the above definition does not depend on the choice of $t\in I$. Suppose
that $\Omega_{\pm}$ are the Fock vacua for the states $\omega_{\pm}$, so that
$\omega_{\pm}=(\Omega_{\pm}|\cdot\Omega_{\pm})$.

Above we parametrized smeared-out fields by solutions to the Klein-Gordon
equation. One usually prefers to parametrize them by space-time functions. We
will distinguish these two parametrizations by using different kinds of
brackets:

$\displaystyle\psi^{*}[f]$
$\displaystyle:=\psi^{*}\Big{(}\frac{1}{\sqrt{2}}G^{\mathrm{PJ}}f\Big{)},$
(8.30a) $\displaystyle\psi[f]$
$\displaystyle:=\psi\Big{(}\frac{1}{\sqrt{2}}G^{\mathrm{PJ}}f\Big{)},\quad
f\in C_{\mathrm{c}}^{\infty}(M).$ (8.30b)

The following theorem gives the interpretation of various propagators in terms
of quantum fields:

######  Theorem 8.4.

The commutator of fields can be expressed by the so-called _Peierls bracket_.

$\displaystyle\big{[}\psi[f_{1}],\psi^{*}[f_{2}]\big{]}$
$\displaystyle=\iint\overline{f_{1}(x)}G^{\mathrm{PJ}}(x,y)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.31a) The vacuum expectation values of the products of fields are expressed
by positive/negative frequency bisolutions:
$\displaystyle\big{(}\Omega_{\pm}|\psi[f_{1}]\psi^{*}[f_{2}]\Omega_{\pm}\big{)}$
$\displaystyle=\iint\overline{f_{1}(x)}G_{\pm}^{(+)}(x,y)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.31b)
$\displaystyle\big{(}\Omega_{\pm}|\psi^{*}[f_{2}]\psi[f_{1}]\Omega_{\pm}\big{)}$
$\displaystyle=\iint\overline{f_{1}(x)}G_{\pm}^{(-)}(x,y)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y.$
(8.31c) Let $\mathrm{T}\\{\\}$ denote the time ordered product and and
$\overline{\mathrm{T}}\\{\\}$ the reverse time ordered product. Assume in
addition the Shale condition for $\omega_{+}$ and $\omega_{-}$. Then the
vacuum expectation values of the time-ordered and reverse time order products
divided by the ovelap of the vacua is expressed by the Feynman, resp. anti-
Feynman inverses: $\displaystyle\frac{\big{(}\Omega_{+}\mkern
2.0mu\big{|}\mkern
2.0mu\mathopen{}\mathrm{T}\\{\psi[f_{1}]\psi^{*}[f_{2}]\\}\Omega_{-}\big{)}}{(\Omega_{+}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})}$
$\displaystyle=\iint\overline{f_{1}(x)}G^{(+-)}(t,s)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.31d) $\displaystyle\frac{\big{(}\Omega_{+}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\overline{\mathrm{T}}\\{\psi[f_{1}]\psi^{*}[f_{2}]\\}\Omega_{-}\big{)}}{(\Omega_{+}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})}$
$\displaystyle=\iint\overline{f_{1}(x)}G^{(-+)}(t,s)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y.$
(8.31e)

The proof of the above theorem is straightforward. In particular, to show
(8.31a) we use (8.24). In the proof of (8.31d) and (8.31e) we use (7.15).

######  Remark 8.5.

Note that the Shale condition for $\omega_{+}$ and $\omega_{-}$ is needed to
write the left hand side of (8.31d) and (8.31e), so that we can use
$\Omega_{+}$ and $\Omega_{-}$ in the same representation. However, the right-
hand sides do not need the Shale condition. So in some sense these right-hand
sides can be used to “define” (or “renormalize”) the left-hand sides!

######  Remark 8.6.

In most physics literature one uses pointlike fields, denoted typically
$\psi^{*}(x),\psi(x)$, $x\in M$, (not to be confused with the fields
$\psi^{*}(u),\psi(u)$ parametrized by $u\in\mathrm{Sol}$, introduced in
(8.28)). Formally, the smeared-out fields are given by

$\displaystyle\psi^{*}[f]$
$\displaystyle:=\int\overline{f(x)}\psi^{*}(x)\mathop{}\\!\mathrm{d}x,$ (8.32)
$\displaystyle\psi[f]$ $\displaystyle:=\int
f(x)\psi(x)\mathop{}\\!\mathrm{d}x.$ (8.33)

Smeared-out fields are more typical for the mathematics literature, since they
can be interpreted as closed densely defined operators (at least in a linear
QFT). Nevertheless, pointlike fields are convenient. We used them in the
introduction. Note that identities (8.31a), (8.31b), (8.31c), (8.31d) and
(8.31e) are equivalent to identities (1.3b), (1.10a), (1.10b), (1.11a) and
(1.11b).

###  8.8 Neutral quantum fields

Suppose that the electromagnetic potential $[A^{\mu}]$ is absent. Then the
Klein-Gordon operator

$\displaystyle K$ $\displaystyle\coloneqq\lvert
g\rvert^{-\frac{1}{4}}\partial_{\mu}\lvert
g\rvert^{\frac{1}{2}}g^{\mu\nu}\partial_{\nu}\lvert g\rvert^{-\frac{1}{4}}-Y.$
(8.34)

is real. In particular, the spaces ${\mathcal{K}}_{\beta}$,
${\mathcal{W}}_{\lambda}$ are equipped with the usual complex conjugation. We
can define their real subspaces ${\mathcal{K}}_{\beta,\mathbb{R}}$,
${\mathcal{W}}_{\lambda,\mathbb{R}}$. Note that ${\mathcal{W}}_{0,\mathbb{R}}$
is a real Krein space equipped with the symplectic form

$\langle v|\omega w\rangle:=\operatorname{Im}(v|Qw).$ (8.35)

The evolution satisfies $R(t,s)=\overline{R(t,s)}$. So it can be restricted to
${\mathcal{W}}_{0,\mathbb{R}}$. Let $\mathrm{Sol}_{\mathbb{R}}$ denote the
space of solutions to the Klein-Gordon equation in
${\mathcal{W}}_{0,\mathbb{R}}$. $\mathrm{Sol}_{\mathbb{R}}$ inherits the real
Krein structure from ${\mathcal{W}}_{0,\mathbb{R}}$.

Let us apply the formalism of the neutral quantization, as described in
Subsection 6.1, to the symplectic space ${\mathcal{W}}_{0,\mathbb{R}}$. Thus
we obtain a linear map $\mathrm{Sol}_{\mathbb{R}}\ni
v\mapsto\phi(v)=\phi(v)^{*}$, satisfying

$\bigl{[}\phi(v),\phi(w)\bigr{]}=\mathrm{i}\langle v|\omega w\rangle,\quad
v,w\in{\mathcal{W}}_{\mathbb{R}}.$ (8.36)

$\phi(v)$ are called smeared-out neutral quantum fields.

We fix all the data of Subsect. 8.6. We define the Fock states $\omega_{\pm}$
by specifying:

$\displaystyle\omega_{\pm}\bigl{(}\phi(v)\phi(w)\bigr{)}$
$\displaystyle=\big{(}\gamma_{t}(v)\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}Q\Pi_{\pm}^{(+)}(t)\gamma_{t}(w)\big{)}.$

Introduce the Fock vacua $\Omega_{\pm}$, so that
$\omega_{\pm}=(\Omega_{\pm}|\cdot\Omega_{\pm})$.

Let us parametrize the smeared-out fields by space-time functions:

$\displaystyle\phi[f]$ $\displaystyle:=\phi(G^{\mathrm{PJ}}f),\quad f\in
C_{\mathrm{c}}^{\infty}(M,\mathbb{R}).$ (8.37)

The following theorem is fully analogous to Thm 8.31.

######  Theorem 8.7.

With conditions analgous to those stated in Thm 8.31, the following identities
are true:

$\displaystyle\big{[}\phi[f_{1}],\phi[f_{2}]\big{]}$ $\displaystyle=\iint
f_{1}(x)G^{\mathrm{PJ}}(x,y)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.38a)
$\displaystyle\big{(}\Omega_{\pm}|\phi[f_{1}]\phi[f_{2}]\Omega_{\pm}\big{)}$
$\displaystyle=\iint
f_{1}(x)G_{\pm}^{(+)}(x,y)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.38b) $\displaystyle\frac{\big{(}\Omega_{+}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\mathrm{T}\\{\phi[f_{1}]\phi[f_{2}]\\}\Omega_{-}\big{)}}{(\Omega_{+}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})}$ $\displaystyle=\iint
f_{1}(x)G^{(+-)}(t,s)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.38c) $\displaystyle\frac{\big{(}\Omega_{+}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\overline{\mathrm{T}}\\{\phi[f_{1}]\phi[f_{2}]\\}\Omega_{-}\big{)}}{(\Omega_{+}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})}$ $\displaystyle=\iint
f_{1}(x)G^{(-+)}(t,s)f_{2}(y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$
(8.38d)

######  Remark 8.8.

In terms of pointlike fields, denoted typically $\phi(x)$, $x\in M$, the
smeared-out fields are given by

$\displaystyle\phi[f]$ $\displaystyle:=\int
f(x)\phi(x)\mathop{}\\!\mathrm{d}x.$ (8.39)

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