# An Evolution Equation Approach to Linear Quantum Field Theory

Jan Dereziński1 Daniel Siemssen2

###### Abstract

In the first part of our paper we analyze bisolutions and inverses of (non-
autonomous) evolution equations. We are mostly interested in pseudo-unitary
evolutions on Krein spaces, which naturally arise in linear Quantum Field
Theory. We prove that with boundary conditions given by a maximal positive and
maximal negative space we can associate an inverse, which can be viewed as a
generalization of the usual Feynman propagator.

In the context of globally hyperbolic manifolds, the Feynman propagator turns
out to be a distinguished inverse of the Klein-Gordon operator. Within the
formalism of Quantum Field Theory on curved spacetimes, the Feynman propagator
yields the expectation values of time-ordered products of fields between the
in and out ‘vacuum’ – the basic ingredient for Feynman diagrams.

##  1 Introduction

The wave equation or, more generally, the Klein-Gordon equation on curved
Lorentzian manifolds is one of classic topics of linear partial differential
equations [22, 14, 1]. One could expect that it is difficult to find new
important concepts in this subject. However, the present paper introduces a
pair of objects associated to Klein-Gordon equation, which we believe are
rather fundamental and to a large extent have been overlooked in the
mathematical literature: the Feynman and anti-Feynman inverse (called also the
Feynman and anti-Feynman propagator).

Quantum Field Theory (QFT) on curved spacetimes is another classic subject,
less known than the wave equation, however also possessing a large literature
[4, 2, 18, 19, 20, 17]. In this literature linear QFT is usually treated as
well understood and the main effort is devoted to define renormalized products
of fields in interacting theories, mostly as formal power series in the
coupling constant. It is often overlooked that linear QFT and its (often
infinite) renormalization can be studied nonperturbatively, without using
formal power series. The Feynman and anti-Feynman inverses play there a
central role. In fact, when we compute the scattering operator by expanding it
into Feynman diagrams, the Feynman propagator is associated to each internal
line.

###  1.1 Propagators in QFT on the Minkowski spacetime

As indicated above, the main motivation of our paper comes from QFT on curved
spacetimes. However, let us first sketch elements of the standard formalism of
QFT on the flat Minkowski spacetime.

The free bosonic field satisfies the Klein–Gordon equation111In the
introduction we use the so-called charged (or complex) formalism for quantum
fields, denoted $\psi(x)$. Note that $\psi(x)$ and $\psi^{*}(x)$ are different
fields. In the neutral (or real) formalism, which is more common in the
mathematics literature, the standard symbol is $\phi(x)$ and the field is
assumed to be self-adjoint: $\phi(x)=\phi^{*}(x)$.

$(-\Box+m^{2})\psi(x)=0.$ (1.1)

To study the field $\psi(x)$ one introduces various “propagators” or “two-
point functions”. Let us list the basic ones, known from many textbooks, e.g.
[3]:

•the forward/backward or retarded/advanced propagator $\displaystyle
G^{\vee/\wedge}(x,y)$
$\displaystyle\coloneqq\frac{1}{(2\pi)^{4}}\int\frac{\mathrm{e}^{-\mathrm{i}(x-y)\cdot
p}}{p^{2}+m^{2}\pm\mathrm{i}0\operatorname{sgn}(p_{0})}\mathop{}\\!\mathrm{d}p,$
(1.2b) •the Feynman/anti-Feynman propagator $\displaystyle
G^{\mathrm{F}/\overline{\mathrm{F}}}(x,y)$
$\displaystyle\coloneqq\frac{1}{(2\pi)^{4}}\int\frac{\mathrm{e}^{-\mathrm{i}(x-y)\cdot
p}}{p^{2}+m^{2}\mp\mathrm{i}0}\mathop{}\\!\mathrm{d}p,$ (1.2c) •the
Pauli–Jordan propagator $\displaystyle G^{\mathrm{PJ}}(x,y)$
$\displaystyle\coloneqq\frac{\mathrm{i}}{(2\pi)^{3}}\int\mathrm{e}^{-\mathrm{i}(x-y)\cdot
p}\operatorname{sgn}(p_{0})\delta(p^{2}+m^{2})\mathop{}\\!\mathrm{d}p,$ (1.2d)
•the positive/negative frequency or particle/antiparticle bisolution
$\displaystyle G^{(\pm)}(x,y)$
$\displaystyle\coloneqq\frac{1}{(2\pi)^{3}}\int\mathrm{e}^{-\mathrm{i}(x-y)\cdot
p}\theta(\pm p_{0})\delta(p^{2}+m^{2})\mathop{}\\!\mathrm{d}p.$ (1.2e)

Mathematically, (1.2b), (1.2c) are distinguished inverses and (1.2d), (1.2e)
are distinguished bisolutions of the Klein–Gordon operator $-\Box+m^{2}$. We
will call them jointly “propagators”.

The propagators express various quantities in QFT. Let us list the most
important ones:

•the commutation relations $\displaystyle[\psi(x),\psi^{*}(y)]$
$\displaystyle=-\mathrm{i}G^{\mathrm{PJ}}(x,y),$ (1.3b) •the vacuum
expectation of products of fields $\displaystyle(\Omega\mkern 2.0mu|\mkern
2.0mu\mathopen{}\psi(x)\psi^{*}(y)\Omega)$ $\displaystyle=G^{(+)}(x,y),$
(1.3c) $\displaystyle(\Omega\mkern 2.0mu|\mkern
2.0mu\mathopen{}\psi^{*}(x)\psi(y)\Omega)$ $\displaystyle=G^{(-)}(x,y),$
(1.3d) •the vacuum expectation of time ordered products of fields
$\displaystyle\big{(}\Omega\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\mathrm{T}\\{\psi(x)\psi^{*}(y)\\}\Omega\big{)}$
$\displaystyle=-\mathrm{i}G^{\mathrm{F}}(x,y).$ (1.3e) •the vacuum expectation
of reverse time ordered products of fields $\displaystyle\big{(}\Omega\mkern
2.0mu\big{|}\mkern
2.0mu\mathopen{}\overline{\mathrm{T}}\\{\psi(x)\psi^{*}(y)\\}\Omega\big{)}$
$\displaystyle=-\mathrm{i}G^{{\overline{\mathrm{F}}}}(x,y).$ (1.3f)

Note the identities satisfied by the propagators:

$\displaystyle G^{\mathrm{PJ}}$ $\displaystyle=G^{\vee}-G^{\wedge}$ (1.4a)
$\displaystyle=\mathrm{i}G^{(+)}-\mathrm{i}G^{(-)},$ (1.4b) $\displaystyle
G^{\mathrm{F}}-G^{\overline{\mathrm{F}}}$
$\displaystyle=\mathrm{i}G^{(+)}+\mathrm{i}G^{(-)},$ (1.4c) $\displaystyle
G^{\mathrm{F}}+G^{\overline{\mathrm{F}}}$ $\displaystyle=G^{\vee}+G^{\wedge},$
(1.4d) $\displaystyle G^{\mathrm{F}}$
$\displaystyle=\mathrm{i}G^{(+)}+G^{\wedge}=\mathrm{i}G^{(-)}+G^{\vee},$
(1.4e) $\displaystyle G^{\overline{\mathrm{F}}}$
$\displaystyle=-\mathrm{i}G^{(+)}+G^{\vee}=-\mathrm{i}G^{(-)}+G^{\wedge}.$
(1.4f)

###  1.2 Evolution approach to QFT on curved spacetimes

Quantum Field Theory based on (1.1) is very simple. More interesting, but
still linear, is QFT on a curved spacetime in the presence of an external
electromagnetic potential $[A^{\mu}]$ and an external scalar potential $Y$.
Then, (1.1) needs to be replaced by a more general equation, which we still
call the Klein–Gordon equation:

$K\psi\coloneqq\bigl{(}\lvert
g\rvert^{-\frac{1}{2}}(\mathrm{i}\partial_{\mu}+A_{\mu})\lvert
g\rvert^{\frac{1}{2}}g^{\mu\nu}(\mathrm{i}\partial_{\nu}+A_{\nu})+Y\bigr{)}\psi=0.$
(1.5)

Note that the most general second-order Hermitian differential operator on
scalar functions is of the form $K$.

One can also ask about various propagators for (1.5). The answer is rather
straightforward in the stationary case, where one can introduce a time
variable so that the metric and external potentials do not depend on time. It
turns out that all the propagators described in (1.2) have natural
generalizations to the stationary case, see e.g. [6]. The relations (1.3) and
the identities (1.4) still hold.

What about QFT and its propagators in the generic, possibly non-stationary
case? It seems that a minimal physically reasonable assumption is that the
spacetime should be globally hyperbolic. It is well known that under this
assumption the forward, backward and Pauli–Jordan propagators (1.2b), (1.2d)
are still well-defined [1, 7]. These propagators play a fundamental role in
the Cauchy problem. We use the name “classical propagators” as the joint name
for these three propagators. Note that the identity (1.4a) still holds. The
main application of classical propagators in QFT is the generalization of the
identity (1.3b): the Pauli–Jordan propagator is responsible for the
commutation relations of fields.

The situation with the other propagators, (1.2c) and (1.2e), which we call
“non-classical”, is more complicated. In the literature it is often claimed
that it makes no sense to ask for distinguished non-classical propagators on
generic spacetimes. The main message of our paper is that for a large class of
non-stationary spacetimes there exist physically relevant distinguished non-
classical propagators.

Let us identify our spacetime $M$ with $I\times\Sigma$, where $I$ describes
the time and $\Sigma$ is a spacelike Cauchy hypersurface. We consider two
cases:

  1. (1)

the slab geometry, where $I=]t_{-},t_{+}[$ is a finite interval,

  2. (2)

$I=\mathbb{R}$.

In the latter case we assume in addition that the Klein-Gordon equation is
asymptotically stationary in the future and in the past. In our opinion such
systems are natural from the physical point of view – asymptotic stationarity
is a necessary condition to apply the ideas of scattering theory, which is the
main means of extracting useful information from QFT. As we will see, the
asymptotic stationarity also allows us to introduce natural definitions of
distinguished non-classical propagators.

In order to compute non-classical (actually, also classical) propagators it is
useful to convert the Klein-Gordon equation into a 1st order evolution
equation for the Cauchy data. The Cauchy data possess a natural pseudo-unitary
structure, which is preserved by the evolution. Asymptotic stationarity
essentially means that the generator of the dynamics has limits as
$t\to\pm\infty$.

Our construction of propagators is divided into several steps. In Sect. 3 we
study evolutions on Hilbertizable spaces. In Sect. 4 we add the Krein
structure to the picture. In Sect. 5 we consider abstract Klein-Gordon
operators. Finally, in Sect. 8 we apply the above abstract results to the
Klein-Gordon equation (1.5) on a globally hyperbolic manifold.

###  1.3 Pseudo-unitary dynamics

A large part of our paper is devoted to an abstract analysis of a pseudo-
unitary evolution, without a direct reference to a space-time setting. We
actually believe that this analysis has other interesting applications besides
QFT on curved spacetimes. After all, symplectic or, after complexification,
pseudo-unitary structures are rather omnipresent in physics and mathematics.
We will see that even on the abstract level, where instead of spacetime we
have only the time variable, one can define analogs of many objects familiar
to the reader from textbooks on QFT.

Note in parenthesis that we prefer to work in the complex setting instead of
the real setting of a symplectic space, perhaps more common in the literature.
Our dynamics acts in a complex Krein space and preserves the so-called charge
form. If the dynamics commutes with the complex conjugation, e.g. if there are
no electromagnetic potentials, then by restricting our dynamics to the real
space we can go back to the real symplectic setting.

As we mentioned above, the analysis simplifies considerably in the stationary
case. The space of Cauchy data has a natural structure of a Hilbert space
(actually, a scale of Hilbert spaces), and therefore the existence of dynamics
is relatively straightforward. In our paper we show that the strategy that
works in the stationary case can be applied with minor changes to the generic
case. Non-autonomous evolutions lead however both to technical and conceptual
problems.

The first issue is technical: one has to construct the evolution. This is not
as straightforward as in the autonomous case. To define the evolution we use
an old result of Kato [21]. In this approach one assumes that the Cauchy data
are described by a Hilbertizable space, and the generators are self-adjoint
with respect to a certain time-dependent scalar product compatible with the
Hilbertizable structure. This condition is naturally satisfied for pseudo-
unitary generators possessing a positive Hamiltonian. Besides, one needs to
make some technical assumptions, which essentially say that the generator of
the evolution does not vary too much in time, so that it acts all the time in
the same Hilbertizable space. Using the evolution it is easy to define the
analogs of classical propagators in the context of Cauchy data, denoted
$E^{\vee},E^{\wedge},E^{\mathrm{PJ}}$.

More interestingly, the time-dependence of the generator leads also to
conceptual problems. It seems that in general there does not exist a single
distinguished state, which can be used to define a representation of our
theory. To remedy this, we consider systems that are asymptotically stationary
in the past and future. Then one can use the limiting past/future symplectic
generator to define two pairs of bisolutions: the “in” and the “out”
bisolutions. In the context of the Cauchy data they are denoted
$E_{\pm}^{(+)},E_{\pm}^{(-)}$. (The plus/minus in the parentheses correspond
to particles/antiparticles; the plus/minus without parentheses correspond to
the future/past.)

What is even more interesting, in the asymptotically stationary case we also
have a natural Feynman and anti-Feynman propagator, denoted
$E^{\mathrm{F}},E^{{\overline{\mathrm{F}}}}$. (We also use an alternative
notation: $E^{(+-)},E^{(-+)}$). The existence of the Feynman propagator uses
heavily the fact that the dynamics is pseudo-unitary. More precisely, we
assume that the evolution acts on a Krein space, which is a class of
Hilbertizable spaces equipped with a distinguished Hermitian form (which is a
substitute for the symplectic form on complex spaces). The existence of the
natural Feynman propagator under rather general conditions is probably the
main new result of our paper. The main fact used in the construction of the
Feynman inverse is the following property of Krein spaces: every maximal
positive subspace is complementary to every maximal negative subspace.

Let us stress that all propagators that we define are introduced without a
reference to the quantized setting, in an abstract setting of an evolution on
a Krein space. They are distinguished bisolutions or inverses (Green’s
operators) of the evolution equation. They satisfy a number of curious
identities, which greatly simplify in the stationary case. Possibly, some of
these identities are recorded in our paper for the first time in the
literature.

###  1.4 Abstract Klein-Gordon operator

Let $I=]t_{-},t_{+}[$ or $I=\mathbb{R}$. Let $L(t)$ a time-dependent positive
operator on a Hilbert space ${\mathcal{K}}$. By an abstract Klein-Gordon
operator we mean an operator of the form

$K:=\big{(}D_{t}+W^{*}(t)\big{)}\frac{1}{\alpha^{2}(t)}\big{(}D_{t}+W(t)\big{)}-L(t),$
(1.6)

acting on the Hilbert space $L^{2}(I,{\mathcal{K}})\simeq
L^{2}(I)\otimes{\mathcal{K}}$.

In our applications, ${\mathcal{K}}$ is going to be the Hilbert space
$L^{2}(\Sigma)$ (the space of functions on spacelike Cauchy surface), $W(t)$
an operator consisting mostly the $A^{0}$ and $L(t)$ is the magnetic
Schrödinger operator on $\Sigma$. Then $K$ becomes the usual Klein-Gordon
operator (1.5) on the Hilbert space $L^{2}(M)\simeq L^{2}(I)\otimes
L^{2}(\Sigma)$.

The operator (1.6) is second order in $t$ — essentially, a 1-dimensional
magnetic Schrödinger operator with operator-valued potentials. It is natural
to associate with it an evolution of Cauchy data on a certain Krein space.
Using the formalism of Sect. 4 we can construct various propagators in the
Cauchy data formalism

$E^{\vee},E^{\wedge},E^{\mathrm{PJ}},E_{\pm}^{(+)},E_{\pm}^{(-)},E^{\mathrm{F}},E^{{\overline{\mathrm{F}}}}.$
(1.7)

It is easy to pass from the Cauchy data propagators to spacetime propagators

$G^{\vee},G^{\wedge},G^{\mathrm{PJ}},G_{\pm}^{(+)},G_{\pm}^{(-)},G^{\mathrm{F}},G^{{\overline{\mathrm{F}}}}.$
(1.8)

In operator theory or the theory of linear differential equations operators
with a non-empty resolvent set are sometimes called well-posed [11]. The most
important class of well-posed operators are of course self-adjoint operators,
however in general the well-posedness is considered as a desirable property
even for non-self-adjoint operators. Inverses of the Klein Gordon operator are
closely related to well-posed realizations of this operator. The situation is
quite different depending on whether we consider a slab–like spacetime, that
is if the interval $I$ is finite, or whether $I=\mathbb{R}$. In the first
case, there are many boundary conditions for the Klein-Gordon operator, each
associated with the corresponding Feynman-type inverse. In the case
$I=\mathbb{R}$ the situation is more difficult and not quite understood. The
Feynman and anti-Feynman inverses are not bounded operators. We conjecture
that under quite general conditions they are boundary values of a certain
distinguished self-adjoint realization of the abstract Klein-Gordon operator.
We will return to this point in Subsect. 1.7.

###  1.5 Bosonic quantization

If a system has a finite number of degrees of freedom (in other words, the
phase space is finite dimensional) quantization is easy and essentially
unique. It is more complicated when we have an infinite number of degrees of
freedom.

There exist two basic approaches to the quantization of bosonic systems:

  * •

The real or neutral formalism starts from a real space equipped with an
antisymmetric form (called also pre-symplectic) and leads to a self-adjoint
field $\phi$.

  * •

The complex or charged formalism starts from a complex space equipped with a
Hermitian form (called sometimes the charge) and leads to a pair of non-self-
adjoint fields $\psi,\psi^{*}$.

The neutral formalism is in a sense more general, since every charged particle
can be understood as two kinds of neutral particles in a presence of a $U(1)$
symmetry.

Quantization can be divided into two steps:

The first step consists in choosing an algebra of observables satisfying
appropriate canonical commutation relations corresponding to the pre-
symplectic or Hermitian form. This step is straightforward.

The second step involves selecting a representation in a Hilbert space.
Usually this is done by fixing a state on the algebra of observables and going
to the GNS representation. If the state is pure quasi-free (which is the
typical choice) this representation naturally acts on a bosonic Fock space.

If the so-called Shale condition is satisfied, then symplectic (in the real
formalism) or pseudo-unitary (in the complex formalism) maps can be
implemented by unitary operators, which we call Bogoliubov implementers. In
linear QFT such operators are used to describe the evolution and the
scattering operator.

We start with the neutral formalism of quantization in Sect. 6. The charged
formalism is explained in the following Sect. 7. The neutral formalism is well
known and described in numerous texts [5, 1]. The complex formalism is rarely
found in mathematical literature. It contains some interesting points that
distinguish it from the neutral formalism, such as gauge-invariant squeezed
states. Therefore we describe it in detail.

###  1.6 QFT on curved spacetimes

Suppose a globally hyperbolic manifold is equipped with the Klein-Gordon
equation (1.5). It is well-known that the space of its solutions can be
equaipped with a natural charge form, so that after comletion, it becomes a
Krein space. If the electromagnetic potential $A^{\mu}(x)$ is zero, then the
Klein-Gordon equation is real, and we can consider the space of its real
solutions, which can be interpreted as a real Krein space.

We can apply the constructions of Sect. 5 about the abstract Klein-Gordon
operator and define various propagators $G^{\bullet}$. Their integral kernels
$G^{\bullet}(x,y)$ are distributions on $M\times M$ with important physical
meanings.

First of all, we have two distinguished positive/negative frequency
bisolutions – those corresponding to the asymptotic past and future,
$G_{\pm}^{(+)}$, resp. $G_{\pm}^{(-)}$. The identity (1.4b) now splits into
two independent identities

$G^{\mathrm{PJ}}=\mathrm{i}G_{\pm}^{(+)}-\mathrm{i}G_{\pm}^{(-)},$ (1.9)

for the “in” and “out” positive/negative frequency bisolutions. After
quantization, these two-point functions correspond to the “in” and “out Fock
representation”. The identities (1.3c) and (1.3d) split into two identities,
one corresponding to the “in vacuum” and the other to the “out vacuum”:

$\displaystyle(\Omega_{\pm}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\psi(x)\psi^{*}(y)\Omega_{\pm})$
$\displaystyle=G_{\pm}^{(+)}(x,y),$ (1.10a) $\displaystyle(\Omega_{\pm}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\psi^{*}(x)\psi(y)\Omega_{\pm})$
$\displaystyle=G_{\pm}^{(-)}(x,y).$ (1.10b)

Note that the states $\Omega_{\pm}$ are natural from the physical point of
view. Let us make a trivial physical remark: it is convenient for an
experimentalist to use stationary conditions at the beginning of the
experiment, so that the “in state” will be close to the “in vacuum”
$\Omega_{-}$. Similarly, measuring the outcome of the experiment an
experimentalist will be facilitated by stationary conditions in the end. Then
the measurement will involve observables closely related to the “out vacuum”
$\Omega_{+}$.

In a large part of the literature the so-called Hadamard property is
considered to be the main criterion for a physically satisfactory state [23,
13, 26]. Note that if one assumes enough smoothness, then
$\big{(}G_{\pm}^{(+)}(x,y),G_{\pm}^{(-)}(x,y)\big{)}$ satisfy the Hadamard
property. This is a non-trivial fact proven by Gérard–Wrochna [15].

It is less obvious that the Feynman and anti-Feynman propagators also possess
natural generalizations to the asymptotically stationary case. Their
construction is probably the main result of our paper. We call them the in-out
Feynman/anti-Feynman propagators, to distinguish them from some other,
noncanonical proposals, such as those mentioned below in (1.14). The in-out
Feynman propagator describes what in popular science books is expressed as
“particles travelling forward in time and antiparticles travelling backwards
in time”. After quantization the Feynman and anti-Feynman propagators satisfy
slight modifications of the identities (1.3e) and (1.3f):

$\displaystyle\frac{\big{(}\Omega_{+}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\mathrm{T}\\{\psi(x)\psi^{*}(y)\\}\Omega_{-}\big{)}}{(\Omega_{+}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})}$
$\displaystyle=-\mathrm{i}G^{\mathrm{F}}(x,y),$ (1.11a)
$\displaystyle\frac{\big{(}\Omega_{+}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}\overline{\mathrm{T}}\\{\psi(x)\psi^{*}(y)\\}\Omega_{-}\big{)}}{(\Omega_{+}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})}$
$\displaystyle=-\mathrm{i}G^{{\overline{\mathrm{F}}}}(x,y).$ (1.11b)

Note that the identities (1.4c)–(1.4f) no longer hold. (They are still true on
the level of singularities of the respective functions – this, however, will
not be discussed here.)

Strictly speaking, (1.11) are true if the Shale condition for the in and out
states holds, so that the vectors $\Omega_{-}$ and $\Omega_{+}$ belong the
same Hilbert space. If the Shale condition is violated, the left hand sides do
not make sense. However, the right hand sides are well defined.

Similarly, the scattering operator between the “in data” and the “out data” is
ill defined when the Shale condition is violated. However, “the renormalized
scattering operator”, that is, divided by the overlap of the vacua
$(\Omega_{+}\mkern 2.0mu|\mkern 2.0mu\mathopen{}\Omega_{-})$, makes sense as a
quadratic form. It has a clear physical meaning: it can be used to compute
ratios of scattering cross-sections.

The time evolution approach that we adopt has one seeming drawback in the
context of curved spacetimes – it hides the underlying general covariance of
the theory, which is obvious in the formalism based on the Klein–Gordon
equation. It forces us to choose a time function, which to a large extent is
arbitrary. Note however that the time variable is “asymptotically well-
defined” for $t$ going to $\pm\infty$, because of the asymptotic stationarity
assumption. Moreover, in order to define the scattering operator (viewed as
the main goal of QFT) one needs to introduce a time variable. Under quite
general assumptions, the resulting scattering operator should not depend on
the choice of the time variable.

Selecting a time variable makes it easy to consider relatively weak regularity
hypothesis. We do not find this very important, however maybe it is worth
noticing that the results of our paper are relatively strong in this respect.

###  1.7 Self-adjointness of the Klein-Gordon operator

It is easy to see that the Klein-Gordon operator $K$ is Hermitian (symmetric)
on, say, $C_{\mathrm{c}}^{\infty}(M)$. This suggests a natural question: is
$K$ essentially self-adjoint, or at least does it possess a distinguished
self-adjoint extension? We conjecture that the Klein-Gordon operator possesses
a distinguished self-adjoint extension for a large class of asymptotically
stationary spacetimes. Moreover, we expect that

$\displaystyle G^{\mathrm{F}}$ $\displaystyle=\lim_{\epsilon\searrow
0}(K-\mathrm{i}\epsilon)^{-1},$ (1.12a) $\displaystyle
G^{\overline{\mathrm{F}}}$ $\displaystyle=\lim_{\epsilon\searrow
0}(K+\mathrm{i}\epsilon)^{-1},$ (1.12b)

in the sense of appropriate weighted spaces.

The above conjectures are obviously true on the Minkowski space. They also
hold in the stationary case. In the absence of the electrostatic potential
this is straightforward, with the electrostatic potential it requires some
work, see [6]. There exists also an interesting paper by Vasy [25] where all
that is proven for asymptotically Minkowskian spacetimes

The question of the self-adjointness of the Klein-Gordon operator is almost
absent in the mathematical literature. There are probably two reasons for
this. First, it seems difficult. The generic Klein-Gordon operator is
unbounded from below and above, and the positivity is usually the major tool
in functional analysis. The second reason is that this question at the first
sight appears not interesting physically. Indeed, the space $L^{2}(M)$ is not
the space of states of any reasonable quantum system and
$\mathrm{e}^{\mathrm{i}tauK}$ seems to have no meaning as a quantum evolution.

However, in physical literature many researchers tacitly assume that the
Klein-Gordon operator is self-adjoint, see e.g. [24]. The Feynman and anti-
Feynman propagator are important for applications, and formally, assuming
(1.12a) and (1.12b) they can be computed from

$\displaystyle(K\pm\mathrm{i}0)^{-1}=\pm\mathrm{i}\int_{0}^{\infty}\mathrm{e}^{\mp\mathrm{i}tK}\mathop{}\\!\mathrm{d}t,$
(1.13)

Note that (1.13) presupposes that one can interpret $K$ as a self-adjoint
operator, so that $(z-K)^{-1}$ can be defined for $z\not\in{{\mathbb{R}}}$.

However all of this is beyond the scope of our paper. In fact, as the analysis
of our paper shows, the Feynman inverse is well-defined in great generality,
whereas making sense of $K$ as a self-adjoint operator seems to be more
difficult, and perhaps possible only with some restrictions.

###  1.8 Comparison with literature

Oversimplifying and exaggerating, researchers studying QFT on curved
spacetimes can be roughly divided into two communities – let us call them
_Hadamardists_ and _Feynmanists_.

The Hadamardists, mostly having a mathematical or General Relativity
background, stress that it is impossible to choose a distinguished state on
the algebra of observables in a locally covariant way, see e.g. [12]. They
argue that one has a lot of freedom in choosing a state, the only restriction
is that it should have the so-called _Hadamard property_ , a certain condition
of their _wave front set_ [23]. They also stress that QFT should be considered
in a local fashion, often restricting attention to a small causally convex
region of a spacetime.

The background of the Feynmanists is usually less mathematical. Their main
goal seems to be to compute scattering amplitudes, cross sections, etc., see
e.g. [8]. Their usual tool is the path integral. Clearly, in this case it is
indispensable to look at the spacetime globally, so that one can define an in
and out state, and the assumption of asymptotic stationarity in the past and
future is rather natural. There is no need to worry about the Hadamard
property – as we mentioned above – it is automatic under rather weak
assumptions thanks to the result of Gérard–Wrochna [15].

There is no contradiction between these two approaches and both have their
philosophical merits. Our paper clearly belongs to the latter approach (in
spite of being rather abstract mathematically).

The basic formalism of symplectic evolution equations that we describe is of
course used in numerous works on QFT. Surprisingly, however, this point of
view is rarely adopted in mathematical literature about QFT on curved
spacetimes, and if so, it is treated as a technical tool. The (trivial)
observation about the existence of two distinguished states on asymptotically
stationary spacetimes seems to be rarely articulated. It is rather obvious
that these are the preferred states for many actual applications, such as the
calculation of the scattering operator.

The in-out Feynman inverse is essentially absent from the mathematical
literature, with recent exceptions of [16, 25]. Sometimes one considers
another generalization of the Feynman propagator: given a Hadamard state with
the two-point functions $G_{\bullet}^{(+/-)}$, one can introduce

$G_{\bullet}^{\mathrm{F}}=\mathrm{i}G_{\bullet}^{(+)}+G^{\wedge}=-\mathrm{i}G_{\bullet}^{(-)}+G^{\vee},$
(1.14)

which is an inverse of the Klein Gordon operator, and can be called the
_Feynman inverse associated to the 2-point function $G_{\bullet}^{(+)}$_.
Note, however, that $G_{\bullet}^{\mathrm{F}}$ is non-unique and, more
importantly, does not satisfy the relation (1.11a), which can be used as the
basis for perturbative calculations of the scattering operator. In the well-
known paper [10] Duistermat and Hörmander prove the existence of the Feynman
parametrix, which is unique only up to a smoothing operator.

In the more physically oriented literature the in-out Feynman operator is
ubiquitous, even if implicitly. It essentially appears each time when the
functional integration method is applied to compute scattering amplitudes.

##  2 Preliminaries

In this section we collect various basic mathematical definitions and facts
that are useful in our paper. Most readers will find them rather obvious –
nevertheless, they should be recorded.

If $A$ is an operator, then $\operatorname{{\mathcal{R}}}(A)$,
$\operatorname{{\mathcal{N}}}(A)$, ${\mathcal{D}}(A)$ and $\sigma(A)$ denote
the range, the nullspace, the domain and the spectrum of $A$.
$B({\mathcal{W}})$ denotes the space of bounded operators on a Banach space
${\mathcal{W}}$.

###  2.1 Scales of Hilbert spaces

Suppose that ${\mathcal{W}}$ is a Hilbert space and $A$ a positive invertible
operator on ${\mathcal{W}}$. Then one defines $A^{-\alpha}{\mathcal{W}}$ as
the domain of $A^{\alpha}$ for $\alpha\geq 0$ and as its anti-dual for
$\alpha<0$. We thus obtain a _scale of nested Hilbert spaces_
$A^{-\alpha}{\mathcal{W}}$, $\alpha\in{{\mathbb{R}}}$, with
$A^{0}{\mathcal{W}}={\mathcal{W}}$ and $A^{-\alpha}{\mathcal{W}}$ continuously
and densely contained in $A^{-\beta}{\mathcal{W}}$ for $\alpha\geq\beta$. By
restriction/extension, the operator $A^{\beta}$ can be interpreted as a
unitary from $A^{-\alpha}{\mathcal{W}}$ to $A^{-\alpha+\beta}{\mathcal{W}}$.
Often we simplify notation by writing ${\mathcal{W}}_{\alpha}$ for
$A^{-\alpha}{\mathcal{W}}$, so that ${\mathcal{W}}_{0}={\mathcal{W}}$ and
${\mathcal{W}}_{1}=\operatorname{{\mathcal{D}}}(A)$.

In practice, the starting point of a construction of a scale of Hilbert spaces
is often not an operator $A$ but a nested pair of Hilbert spaces. More
precisely, suppose that $({\mathcal{W}},{\mathcal{V}})$ is a pair of Hilbert
spaces, where ${\mathcal{V}}$ is densely and continuously embedded in
${\mathcal{W}}$. Then there exists a unique invertible positive self-adjoint
operator $A$ on ${\mathcal{W}}$ with the domain ${\mathcal{V}}$ such that

$(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}v)_{{\mathcal{V}}}=(Av\mkern
2.0mu|\mkern 2.0mu\mathopen{}Av)_{{\mathcal{W}}},\quad v\in{\mathcal{V}}.$
(2.1)

We can then define the scale
${\mathcal{W}}_{\alpha}=A^{-\alpha}{\mathcal{W}},\alpha\in{{\mathbb{R}}}$.
Note that ${\mathcal{W}}={\mathcal{W}}_{0}$ and
${\mathcal{V}}={\mathcal{W}}_{1}$.

In the above setup ${\mathcal{W}}={\mathcal{W}}_{0}$ is treated as the central
element of the space, its “pivot”. Often it is natural to shift the pivot by
$\frac{1}{2}$ and set
${\mathcal{W}}_{\alpha}=A^{-\frac{1}{2}-\alpha}{\mathcal{W}}$, so that
${\mathcal{V}}={\mathcal{W}}_{\frac{1}{2}}$ is the form domain of $A$ viewed
as an operator on ${\mathcal{W}}_{0}$ and
${\mathcal{W}}={\mathcal{W}}_{-\frac{1}{2}}$ is the anti-dual of the form
domain.

Note the following facts, which we will often use:

######  Proposition 2.1.

  1. (1)

$A\in B({\mathcal{K}}_{\alpha},{\mathcal{K}}_{\beta})$ implies $A^{*}\in
B({\mathcal{K}}_{-\beta},{\mathcal{K}}_{-\alpha})$.

  2. (2)

If $A\in B({\mathcal{K}}_{\alpha_{0}},{\mathcal{K}}_{\beta_{0}})$ restricts to
$A\in B({\mathcal{K}}_{\alpha_{1}},{\mathcal{K}}_{\beta_{1}})$, then restricts
to
$B\in({\mathcal{K}}_{(1-\tau)\alpha_{0}+\tau\alpha_{1}},{\mathcal{K}}_{(1-\tau)\beta_{0}+\tau\beta_{1}})$
for $\tau\in[0,1]$.

###  2.2 One-parameter groups

Let ${\mathcal{W}}$ be a Banach space. Recall that a _one-parameter group on
${\mathcal{W}}$_ is a homomorphism

${{\mathbb{R}}}\ni t\mapsto R(t)\in B({\mathcal{W}}).$ (2.2)

It is well known that to every strongly continuous one-parameter group $R(t)$
one can uniquely associate a densely defined operator $B$ called the
_generator of $R(t)$_. One then writes $R(t)=\mathrm{e}^{-\mathrm{i}tB}$. It
can be shown that $\operatorname{{\mathcal{D}}}(B)$ is preserved by $R(t)$ and
the following equation is true:

$(\partial_{t}+\mathrm{i}B)R(t)w=0,\quad w\in\operatorname{{\mathcal{D}}}(B).$
(2.3)

Let us now specialize to the case when is ${\mathcal{W}}$ a Hilbert space. A
unitary group on ${\mathcal{W}}$ is always of the form
$\mathrm{e}^{-\mathrm{i}tB}$, where $B$ is a self-adjoint operator. Let
$\langle B\rangle\coloneqq(B^{2}+1)^{\frac{1}{2}}$. Note that
$\mathrm{e}^{-\mathrm{i}tB}$ preserves the scale
${\mathcal{W}}_{\alpha}\coloneqq\langle B\rangle^{-\alpha}{\mathcal{W}}$ for
$\alpha\geq 0$, and can be uniquely extended by continuity to
${\mathcal{W}}_{\alpha}$ for $\alpha\leq 0$. For any $\alpha$ we have

$(\partial_{t}+\mathrm{i}B)\mathrm{e}^{-\mathrm{i}tB}w=0,\quad
w\in{\mathcal{W}}_{1+\alpha},$ (2.4)

where the left-hand side of (2.4) is understood as an element of
${\mathcal{W}}_{\alpha}$.

In practice, two choices of $\alpha$ are especially useful: $\alpha=0$
corresponds precisely to (2.3), and $\alpha=-\frac{1}{2}$ means that (2.4) is
considered on the form domain of $B$. We will use both points of view when
considering a natural setup for non-autonomous evolutions, see Thm. 3.11.

If in addition $B$ is invertible, then we can slightly modify the scale of
Hilbert spaces by setting ${\mathcal{W}}_{\alpha}\coloneqq\lvert
B\rvert^{-\alpha}{\mathcal{W}}$. Note that $B$ is then unitary from
${\mathcal{W}}_{\alpha}$ to ${\mathcal{W}}_{\alpha-1}$.

###  2.3 Hilbertizable spaces

######  Definition 2.2.

Let ${\mathcal{W}}$ be a complex222Analogous definitions and results are valid
for _real_ Hilbertizable spaces. topological vector space. We say that it is
_Hilbertizable_ if it has a topology of a Hilbert space for some scalar
product $(\cdot\mkern 2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{\bullet}$ on
${\mathcal{W}}$. We will then say that $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{\bullet}$ is _compatible with (the Hilbertizable
structure of)_ ${\mathcal{W}}$. The subscript $\bullet$ serves as a
placeholder for a name of a scalar product. The Hilbert space
$\bigl{(}{\mathcal{W}},(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{\bullet}\bigr{)}$ will be occasionally denoted
${\mathcal{W}}_{\bullet}$. The corresponding norm will be denoted
$\lVert\,\cdot\,\rVert_{\bullet}$.

In what follows ${\mathcal{W}}$ is a Hilbertizable space. Let $(\cdot\mkern
2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{1}$, $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{2}$ be two scalar products compatible with
${\mathcal{W}}$. Then there exist constants $0<c\leq C$ such that

$c(w\mkern 2.0mu|\mkern 2.0mu\mathopen{}w)_{1}\leq(w\mkern 2.0mu|\mkern
2.0mu\mathopen{}w)_{2}\leq C(w\mkern 2.0mu|\mkern 2.0mu\mathopen{}w)_{1}.$
(2.5)

Let $R$ be a linear operator on ${\mathcal{W}}$. We say that it is _bounded_
if for some (hence for all) compatible scalar products $(\cdot\mkern
2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{\bullet}$ there exists a constant
$C_{\bullet}$ such that

$\lVert Rw\rVert_{\bullet}\leq C_{\bullet}\lVert w\rVert_{\bullet}.$ (2.6)

Suppose that $A$ is a (densely defined) operator on ${\mathcal{W}}$. We say
that it is _similar to self-adjoint_ if there exists a compatible scalar
product $(\cdot\mkern 2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{\bullet}$ such that
$A$ is self-adjoint with respect to $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{\bullet}$. Note that for such operators the spectral
theorem applies. In particular, for any (complex-valued) Borel function $f$ on
the spectrum of $A$ we can define $f(A)$.

Let $Q$ be a sesquilinear form on ${\mathcal{W}}$. We say that it is _bounded_
if for some (hence for all) compatible scalar products $(\cdot\mkern
2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{\bullet}$ there exists $C_{\bullet}$ such
that

$\lvert(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)\rvert\leq C_{\bullet}\lVert
v\rVert_{\bullet}\lVert w\rVert_{\bullet},\quad v,w\in{\mathcal{W}}.$ (2.7)

Note that on Hilbertizable spaces we do not have a natural identification of
sesquilinear forms with operators.

###  2.4 Interpolation between Hilbertizable spaces

######  Definition 2.3.

A pair of Hilbertizable spaces $({\mathcal{W}},{\mathcal{V}})$, where
${\mathcal{V}}$ is densely and continuously embedded in ${\mathcal{W}}$, will
be called a _nested Hilbertizable pair_.

After fixing scalar products $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{{\mathcal{V}},\bullet}$ and $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{{\mathcal{W}},\bullet}$ compatible with
${\mathcal{V}}$, resp. ${\mathcal{W}}$, we can interpolate between the Hilbert
spaces ${\mathcal{V}}_{\bullet}$ and ${\mathcal{W}}_{\bullet}$ obtaining a
scale of Hilbert spaces ${\mathcal{W}}_{\alpha,\bullet}$,
$\alpha\in{{\mathbb{R}}}$, with
${\mathcal{V}}_{\bullet}={\mathcal{W}}_{1,\bullet}$ and
${\mathcal{W}}={\mathcal{W}}_{0,\bullet}$. By the Heinz–Kato Theorem, for
$\alpha\in[0,1]$ they do not depend on the choice of scalar products
$(\cdot\mkern 2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{{\mathcal{W}},\bullet}$ and
$(\cdot\mkern 2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{{\mathcal{V}},\bullet}$ as
Hilbertizable spaces. Therefore, the family of Hilbertizable spaces
${\mathcal{W}}_{\alpha}$, $\alpha\in[0,1]$, is uniquely defined.

If $R\in B({\mathcal{W}})$ and its restriction to ${\mathcal{V}}$ belongs to
$B({\mathcal{V}})$, then $R$ restricts to $B({\mathcal{W}}_{\alpha})$ for
$0\leq\alpha\leq 1$.

###  2.5 From complex to real spaces and back

To pass from a complex space to a real one, it is useful to have the notion of
a conjugation:

Let ${\mathcal{W}}$ be a complex space. An antilinear involution
$v\mapsto\overline{v}$ on ${\mathcal{W}}$ will be called a _conjugation_. In
the context of Hilbertizable spaces we always assume that conjugations are
bounded. For an operator $R$ on ${\mathcal{W}}$ we set

$\overline{R}v\coloneqq\overline{R\overline{v}},\quad
R^{\\#}\coloneqq\overline{R}^{*}.$ (2.8)

If $R$ satisfies $\overline{R}=\pm R$, it will be called _real_ resp.  _anti-
real_. The _real subspace of ${\mathcal{W}}$_ is defined as

${\mathcal{W}}_{{\mathbb{R}}}\coloneqq\\{w\in{\mathcal{W}}\ :\
\overline{w}=w\\}.$ (2.9)

Conversely, to pass from a real space to a complex one, suppose now that
${\mathcal{Y}}$ is a real space. Then
${\mathcal{Y}}\otimes_{{\mathbb{R}}}{{\mathbb{C}}}={{\mathbb{C}}}{\mathcal{Y}}$
will denote the _complexification_ of ${\mathcal{Y}}$ (i.e., for every
$w\in{\mathcal{W}}$ we can write $w=w_{R}+\mathrm{i}w_{I}$ with
$w_{R},w_{I}\in{\mathcal{Y}}$), and we have the natural conjugation
$\overline{v_{R}+\mathrm{i}v_{I}}=v_{R}-\mathrm{i}v_{I}$.

###  2.6 Complexification of (anti-)symmetric forms

Let ${\mathcal{Y}}$ be a real space.

Every symmetric form $q$ on ${\mathcal{Y}}$, and thus in particular every
scalar product, extends to a Hermitian form on ${{\mathbb{C}}}{\mathcal{Y}}$:

$\begin{split}\big{(}v_{R}+\mathrm{i}v_{I}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}Q(w_{R}+\mathrm{i}w_{I})\big{)}&\coloneqq\langle v_{R}\mkern
2.0mu|\mkern 2.0mu\mathopen{}qw_{R}\rangle+\langle v_{I}\mkern 2.0mu|\mkern
2.0mu\mathopen{}qw_{I}\rangle\\\ &\quad-\mathrm{i}\langle v_{I}\mkern
2.0mu|\mkern 2.0mu\mathopen{}qw_{R}\rangle+\mathrm{i}\langle v_{R}\mkern
2.0mu|\mkern 2.0mu\mathopen{}qw_{I}\rangle.\end{split}$ (2.10)

Note the additional property $\overline{(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw)}=(\overline{v}\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q\overline{w})$.

Extending an antisymmetric form $\omega$ on ${\mathcal{Y}}$ to a Hermitian
form on ${{\mathbb{C}}}{\mathcal{Y}}$ is slightly different:

$\begin{split}\big{(}v_{R}+\mathrm{i}v_{I}\mkern 2.0mu\big{|}\mkern
2.0mu\mathopen{}Q(w_{R}+\mathrm{i}w_{I})\big{)}&\coloneqq\langle v_{I}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\omega w_{R}\rangle-\langle v_{R}\mkern
2.0mu|\mkern 2.0mu\mathopen{}\omega w_{I}\rangle\\\ &\quad+\mathrm{i}\langle
v_{R}\mkern 2.0mu|\mkern 2.0mu\mathopen{}\omega w_{R}\rangle+\mathrm{i}\langle
v_{I}\mkern 2.0mu|\mkern 2.0mu\mathopen{}\omega w_{I}\rangle.\end{split}$
(2.11)

Note the additional property $\overline{(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw)}=-(\overline{v}\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q\overline{w})$, which also differs from the symmetric case
above.

###  2.7 Realification of Hermitian forms

Let $Q$ be a Hermitian form on ${\mathcal{W}}$.

We say that a conjugation $\overline{\,\cdot\vphantom{1}\,}$ _preserves_ $Q$
if

$\overline{(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)}=(\overline{v}\mkern
2.0mu|\mkern 2.0mu\mathopen{}Q\overline{w}).$ (2.12)

In that case,

$\operatorname{Re}{}(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw),\quad
v,w\in{\mathcal{W}}_{{\mathbb{R}}},$ (2.13)

is a symmetric form on ${\mathcal{W}}_{{\mathbb{R}}}$. Note that
$\operatorname{Im}{}(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)=0$ on
${\mathcal{W}}_{{\mathbb{R}}}$.

Similarly, we say that a conjugation $\overline{\,\cdot\vphantom{1}\,}$ _anti-
preserves_ $Q$ if

$\overline{(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)}=-(\overline{v}\mkern
2.0mu|\mkern 2.0mu\mathopen{}Q\overline{w}).$ (2.14)

In that case,

$\operatorname{Im}{}(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw),\quad
v,w\in{\mathcal{W}}_{{\mathbb{R}}},$ (2.15)

is an antisymmetric form on ${\mathcal{W}}_{{\mathbb{R}}}$. Note that
$\operatorname{Re}{}(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)=0$ on
${\mathcal{W}}_{{\mathbb{R}}}$.

###  2.8 Involutions

######  Definition 2.4.

We say that a pair
$({\mathcal{Z}}_{\bullet}^{(+)},{\mathcal{Z}}_{\bullet}^{(-)})$ of subspaces
of a vector space ${\mathcal{W}}$ is _complementary_ if

${\mathcal{Z}}_{\bullet}^{(+)}\cap{\mathcal{Z}}_{\bullet}^{(-)}=\\{0\\},\quad{\mathcal{Z}}_{\bullet}^{(+)}+{\mathcal{Z}}_{\bullet}^{(-)}={\mathcal{W}}.$

######  Definition 2.5.

An operator $S_{\bullet}$ on ${\mathcal{W}}$ is called an involution, if
$S_{\bullet}^{2}={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$.

We can associate various objects with $S_{\bullet}$:

$\Pi^{(\pm)}_{\bullet}\coloneqq\frac{1}{2}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\pm
S_{\bullet}),\quad{\mathcal{Z}}^{(\pm)}_{\bullet}\coloneqq\operatorname{{\mathcal{R}}}(\Pi^{(\pm)}_{\bullet}).$
(2.16)

$(\Pi_{\bullet}^{(+)},\Pi_{\bullet}^{(-)})$ is a pair of complementary
projections and
$({\mathcal{Z}}_{\bullet}^{(+)},{\mathcal{Z}}_{\bullet}^{(-)})$ is the
corresponding pair of complementary subspaces.

A possible name for ${\mathcal{Z}}_{\bullet}^{(+)}$ is the _positive space_ ,
and for ${\mathcal{Z}}_{\bullet}^{(-)}$ is the _negative space_ (associated
with $S_{\bullet}$). We will however prefer names suggested by QFT:
${\mathcal{Z}}_{\bullet}^{(+)}$ is the _particle space_ , and
${\mathcal{Z}}_{\bullet}^{(-)}$ the _antiparticle space_.

If ${\mathcal{W}}$ is Hilbertizable, we will usually assume that $S_{\bullet}$
is bounded. Then so are $\Pi_{\bullet}^{(+)}$ and $\Pi_{\bullet}^{(-)}$,
moreover, ${\mathcal{Z}}_{\bullet}^{(+)}$ and ${\mathcal{Z}}_{\bullet}^{(-)}$
are closed.

###  2.9 Pairs of involutions

Suppose that $S_{1}$ and $S_{2}$ are two bounded involutions on
${\mathcal{W}}$. Let

$\displaystyle\Pi_{i}^{(\pm)}:=\frac{1}{2}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\pm
S_{i}),\quad{\mathcal{Z}}_{i}^{(\pm)}:=\operatorname{{\mathcal{R}}}(\Pi_{i}^{(\pm)}),\quad
i=1,2,$ (2.17)

be the corresponding pairs of complementary projections and subspaces. We
define

$\displaystyle\Upsilon$ $\displaystyle\coloneqq{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-(\Pi_{1}^{(+)}-\Pi_{2}^{(+)})^{2}={{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-(\Pi_{1}^{(-)}-\Pi_{2}^{(-)})^{2}$ (2.18a)
$\displaystyle\mathrel{\phantom{\coloneqq}\mathllap{=}}\Pi_{1}^{(+)}\Pi_{2}^{(+)}+\Pi_{2}^{(-)}\Pi_{1}^{(-)}=\Pi_{2}^{(+)}\Pi_{1}^{(+)}+\Pi_{1}^{(-)}\Pi_{2}^{(-)}$
(2.18b)
$\displaystyle\mathrel{\phantom{\coloneqq}\mathllap{=}}(\Pi_{1}^{(+)}+\Pi_{2}^{(-)})(\Pi_{2}^{(+)}+\Pi_{1}^{(-)})=(\Pi_{2}^{(+)}+\Pi_{1}^{(-)})(\Pi_{1}^{(+)}+\Pi_{2}^{(-)})$
(2.18c)
$\displaystyle\mathrel{\phantom{\coloneqq}\mathllap{=}}(\Pi_{1}^{(+)}\Pi_{2}^{(+)}+\Pi_{2}^{(-)})(\Pi_{2}^{(+)}\Pi_{1}^{(+)}+\Pi_{1}^{(-)})=(\Pi_{2}^{(+)}\Pi_{1}^{(+)}+\Pi_{1}^{(-)})(\Pi_{1}^{(+)}\Pi_{2}^{(+)}+\Pi_{2}^{(-)})$
(2.18d)
$\displaystyle\mathrel{\phantom{\coloneqq}\mathllap{=}}\frac{1}{4}(2+S_{1}S_{2}+S_{2}S_{1})=\frac{1}{4}(S_{1}+S_{2})^{2}.$
(2.18e)

Observe that $\Upsilon$ commutes with $\Pi_{1}^{(+)}$, $\Pi_{1}^{(-)}$,
$\Pi_{2}^{(+)}$ and $\Pi_{2}^{(-)}$.

######  Proposition 2.6.

The following are equivalent:

  1. (i)

$\Upsilon$ is invertible.

  2. (ii)

$\Pi_{1}^{(+)}+\Pi_{2}^{(-)}$ and $\Pi_{2}^{(+)}+\Pi_{1}^{(-)}$ are
invertible.

  3. (iii)

$\Pi_{1}^{(+)}\Pi_{2}^{(+)}+\Pi_{2}^{(-)}$ and
$\Pi_{2}^{(+)}\Pi_{1}^{(+)}+\Pi_{1}^{(-)}$ are invertible.

Moreover, if one of the above holds, then the pairs
$({\mathcal{Z}}_{1}^{(+)},{\mathcal{Z}}_{2}^{(-)})$ as well as
$({\mathcal{Z}}_{2}^{(+)},{\mathcal{Z}}_{1}^{(-)})$ are complementary.

###### Proof.

(i)$\iff$(ii) and (i)$\iff$(iii) follow from (2.18c) and (2.18d) by the
following easy fact: If $R,S,T$ are maps such that $R=ST=TS$, then $R$ is
bijective if and only if both $T$ and $S$ are bijective.

The last implication follows from the next proposition. ∎

In the setting of the above propostion we can use $\Upsilon$ to construct two
pairs of complementary projections:

######  Proposition 2.7.

Suppose that $\Upsilon$ is invertible. Then

$\displaystyle\Lambda_{12}^{(+)}$
$\displaystyle\coloneqq\Pi_{1}^{(+)}\Upsilon^{-1}\Pi_{2}^{(+)}$
$\displaystyle\text{is the projection onto ${\mathcal{Z}}_{1}^{(+)}$ along
${\mathcal{Z}}_{2}^{(-)}$},$ $\displaystyle\Lambda_{21}^{(-)}$
$\displaystyle\coloneqq\Pi_{2}^{(-)}\Upsilon^{-1}\Pi_{1}^{(-)}$
$\displaystyle\text{is the projection onto ${\mathcal{Z}}_{2}^{(-)}$ along
${\mathcal{Z}}_{1}^{(+)}$},$ $\displaystyle\Lambda_{21}^{(+)}$
$\displaystyle\coloneqq\Pi_{2}^{(+)}\Upsilon^{-1}\Pi_{1}^{(+)}$
$\displaystyle\text{is the projection onto ${\mathcal{Z}}_{2}^{(+)}$ along
${\mathcal{Z}}_{1}^{(-)}$},$ $\displaystyle\Lambda_{12}^{(-)}$
$\displaystyle\coloneqq\Pi_{1}^{(-)}\Upsilon^{-1}\Pi_{2}^{(-)}$
$\displaystyle\text{is the projection onto ${\mathcal{Z}}_{1}^{(-)}$ along
${\mathcal{Z}}_{2}^{(+)}$}.$

In particular,

$\Lambda_{12}^{(+)}+\Lambda_{21}^{(-)}={{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}},\quad\Lambda_{21}^{(+)}+\Lambda_{12}^{(-)}={{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}.$ (2.19)

###### Proof.

First we check that $\Lambda_{12}^{(+)}$ is a projection:

$\displaystyle\bigl{(}\Lambda_{12}^{(+)}\bigr{)}^{2}$
$\displaystyle=\Pi_{1}^{(+)}\Upsilon^{-1}\Pi_{2}^{(+)}\Pi_{1}^{(+)}\Upsilon^{-1}\Pi_{2}^{(+)}$
$\displaystyle=\Pi_{1}^{(+)}\Upsilon^{-1}(\Pi_{2}^{(+)}\Pi_{1}^{(+)}+\Pi_{1}^{(-)}\Pi_{2}^{(-)})\Upsilon^{-1}\Pi_{2}^{(+)}=\Lambda_{12}^{(+)}.$

Moreover,

$\displaystyle\Lambda_{12}^{(+)}$
$\displaystyle=\Pi_{1}^{(+)}(\Pi_{2}^{(+)}+\Pi_{1}^{(-)})\Upsilon^{-1}=\Upsilon^{-1}(\Pi_{1}^{(+)}+\Pi_{2}^{(-)})\Pi_{2}^{(+)}.$

But $(\Pi_{2}^{(+)}+\Pi_{1}^{(-)})\Upsilon^{-1}$ and
$\Upsilon^{-1}(\Pi_{1}^{(+)}+\Pi_{2}^{(-)})$ are invertible. Hence
$\operatorname{{\mathcal{R}}}(\Lambda_{12}^{(+)})=\operatorname{{\mathcal{R}}}(\Pi_{1}^{(+)})$
and
$\operatorname{{\mathcal{N}}}(\Lambda_{12}^{(+)})=\operatorname{{\mathcal{N}}}(\Pi_{2}^{(+)})=\operatorname{{\mathcal{R}}}(\Pi_{2}^{(-)})$.
This proves the statement of the proposition about $\Lambda_{12}^{(+)}$. The
remaining statements are proven analogously. ∎

##  3 Evolutions on Hilbertizable spaces

In this section we investigate the concept of an evolution (family) in the
Banach space setting and without the additional pseudo-unitary structure,
which will be added in later sections. Already in the present setting we can
try to define abstract versions of “forward/backward”, “Pauli-Jordan”,
“particle/antiparticle”, “Feynman/anti-Feynman” propagators and derive their
basic properties. The existence of the abstract version of the Feynman
propagator (inverse) will depend on a certain property called “asymptotic
complementarity”, which in general is not guaranteed to hold. In the next
section we will see that this property automatically holds under some natural
assumptions typical of QFT.

Throughout the section, $-\infty\leq t_{-}<t_{+}\leq+\infty$. For brevity, we
write $I\coloneqq\mathopen{]}t_{-},t_{+}\mathclose{[}$. Without limiting the
generality, we will always assume that $0\in I$. Moreover, we will always
assume that either $t_{\pm}$ are both finite or both infinite. The case where
only one of $t_{\pm}$ is infinite can be deduced from the other cases.

In typical situations to define an evolution one first introduces a time-
dependent family of operators $t\mapsto-\mathrm{i}B(t)$ that generate
$R(t,s)$. The necessary and sufficient conditions for an operator $B$ to
generate an autonomous evolution or, what is equivalent, a strongly continuous
one-parameter group are well known and relatively simple. In the non-
autonomous case, there exist various relatively complicated theorems
describing sufficient conditions. Unfortunately, it seems that a complete
theory on this subject is not available.

In the first part of this section we will avoid discussing the topic of
generators of a non-autonomous evolutions apart from heuristic remarks. We
will treat the evolution as given. The evolution equation operator
$M:=\partial_{t}+\mathrm{i}B(t)$ will be just a heuristic concept, without a
rigorous meaning. However, “bisolutions” and “inverses of $M$” will be
rigorously defined. They will be the main topic of this section.

In Subsection 3.7 we describe a possible approach to the generation of non-
autonomous evolutions that seems general enough for most applications in a
pseudo-unitary setting.

###  3.1 Concept of an evolution

######  Definition 3.1.

Let ${\mathcal{W}}$ be a Banach space. We say that the two-parameter family

$I\times I\ni(t,s)\mapsto R(t,s)\in B({\mathcal{W}})$ (3.1)

is a _strongly continuous evolution (family) on ${\mathcal{W}}$_ if for all
$r,s,t\in I$, we have the identities

$R(t,t)={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}},\quad R(t,s)R(s,r)=R(t,r),$ (3.2)

and the map (3.1) is strongly continuous.

One can also consider evolutions parametrized by the closed interval
$I^{\mathrm{cl}}\coloneqq[t_{-},t_{+}]$ instead of $I$, with the obvious
changes in the definition.

Note also that Def. 3.1 involves both forward and backward evolution, since we
do not assume $t\geq s$ in $R(t,s)$. In other words, this definition is a
generalization of a one-parameter group instead of a one-parameter semigroup.

###  3.2 Generators of evolution

Until the end of this section we consider a strongly continuous evolution
$R(t,s)$, $t,s\in I$, on a Hilbertizable space ${\mathcal{W}}$.

If $R(t,s)=R(t-s,0)$ for all $t,s,t-s,0\in I$, we say that the evolution is
_autonomous_. An autonomous evolution can always be extended to
${{\mathbb{R}}}$ in the obvious way. Setting $R(t)\coloneqq R(t,0)$, we obtain
a _strongly continuous one-parameter group_. As we have already mentioned, we
can then write $R(t)=\mathrm{e}^{-\mathrm{i}tB}$, where $B$ is a certain
unique, densely defined, closed operator called the _generator of $R$_.

For non-autonomous evolutions, the concept of a generator is understood only
under some special assumptions. Heuristically, the operator-valued function
$I\ni t\mapsto B(t)$ is called the _(time-dependent) generator of $R(t,s)$_ if

$B(t)\coloneqq\bigr{(}\mathrm{i}\partial_{t}R(t,s)\bigl{)}R(s,t).$

Note that the evolution should satisfy in some sense

$\displaystyle\mathrm{i}\partial_{t}R(t,s)v$ $\displaystyle=B(t)R(t,s)v,$
(3.3a) $\displaystyle-\mathrm{i}\partial_{s}R(t,s)v$
$\displaystyle=R(t,s)B(s)v.$ (3.3b)

A possible rigorous meaning of the concept of a time-dependent generator will
be discussed in Subsect. 3.7.

###  3.3 Bisolutions and inverses of the evolution equation

Introducing the (heuristic) generator $B(t)$, we can consider the (still
heuristic) _evolution operator_

$M:=\partial_{t}+\mathrm{i}B(t),$ (3.4)

We will say that an operator $E$ is a _bisolution_ resp. an _inverse or
Green’s operator of $M$_ if it is maps $C_{\mathrm{c}}(I,{\mathcal{W}})\to
C(I,{\mathcal{W}})$ and satisfies (heuristically)

$\displaystyle\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}Ew$
$\displaystyle=0,$ $\displaystyle\quad
E\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}v$ $\displaystyle=0,$ (3.5a)
$\displaystyle\text{resp.}\quad\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}Ew$
$\displaystyle=w,$ $\displaystyle\quad
E\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}v$ $\displaystyle=v.$ (3.5b)

A rigorous version of (3.5a) and (3.5b) will be given in Subsect. 3.8.

Note that all the definitions of operators that we will call inverses and
bisolution will be rigorous. Yet, for the time being, they will satisfy the
conditions (3.5a) or (3.5b) only on a heuristic level.

The following definition introduces the two most natural inverses and the most
natural bisolution:

######  Definition 3.2.

Define the operators $E^{\bullet}:C_{\mathrm{c}}(I,{\mathcal{W}})\to
C(I,{\mathcal{W}})$

$(E^{\bullet}w)(t)\coloneqq\int_{I}E^{\bullet}(t,s)w(s)\mathop{}\\!\mathrm{d}s,\quad\bullet=\mathrm{PJ},\vee,\wedge,$
(3.6)

by their _temporal integral kernels_

$\displaystyle E^{\mathrm{PJ}}(t,s)$ $\displaystyle\coloneqq R(t,s),$ (3.7a)
$\displaystyle E^{\vee}(t,s)$ $\displaystyle\coloneqq\theta(t-s)R(t,s),$
(3.7b) $\displaystyle E^{\wedge}(t,s)$
$\displaystyle\coloneqq-\theta(s-t)R(t,s).$ (3.7c)

$E^{\mathrm{PJ}}$ is called the _Pauli–Jordan bisolution_ and
$E^{\vee},E^{\wedge}$ are called the _forward_ resp.  _backward inverse_.
Jointly, we call them _classical propagators_.

Clearly, we have

$E^{\mathrm{PJ}}=E^{\vee}-E^{\wedge},$ (3.8)

which is analogous to (1.4a).

If $I$ is finite, then the operators $E^{\vee},E^{\wedge},E^{\mathrm{PJ}}$ can
be extended to bounded operators on the Hilbertizable space
$L^{2}(I,{\mathcal{W}})=L^{2}(I)\otimes{\mathcal{W}}$.

If $I=\mathbb{R}$, they cannot be expected to be bounded on
$L^{2}(\mathbb{R},{\mathcal{W}})$. However, if $R(t,s)$ is uniformly bounded
(which we will typically assume), then $E^{\vee},E^{\wedge},E^{\mathrm{PJ}}$
are bounded as operators $\langle
t\rangle^{-s}L^{2}(I,{\mathcal{W}})\to\langle
t\rangle^{s}L^{2}(I,{\mathcal{W}})$ for $s>\frac{1}{2}$.

###  3.4 Inverses associated to boundary conditions

Consider first finite $t_{-},t_{+}$. Let
$({\mathcal{Z}}_{+}^{(+)},{\mathcal{Z}}_{-}^{(-)})$ be a pair of closed
subspaces of ${\mathcal{W}}$, which will define the boundary conditions at
$t_{-}$ and $t_{+}$, respectively. Sometimes we will call
${\mathcal{Z}}_{+}^{(+)}$ the out particle space and ${\mathcal{Z}}_{-}^{(-)}$
the in antiparticle space. We set

$\displaystyle{\mathcal{Z}}_{-}^{(-)}(t)$ $\displaystyle\coloneqq
R(t,t_{-}){\mathcal{Z}}_{-}^{(-)},$ (3.9a)
$\displaystyle{\mathcal{Z}}_{+}^{(+)}(t)$ $\displaystyle\coloneqq
R(t,t_{+}){\mathcal{Z}}_{+}^{(+)}.$ (3.9b)

Note that

$\displaystyle R(t,s){\mathcal{Z}}_{-}^{(-)}(s)$
$\displaystyle={\mathcal{Z}}_{-}^{(-)}(t),$ (3.10a) $\displaystyle
R(t,s){\mathcal{Z}}_{+}^{(+)}(s)$ $\displaystyle={\mathcal{Z}}_{+}^{(+)}(t).$
(3.10b)

If $t_{\pm}=\pm\infty$, we suppose that
$\big{(}{\mathcal{Z}}_{+}^{(+)}(t),{\mathcal{Z}}_{-}^{(-)}(t)\big{)}$ is a
family of pairs of subspaces satisfying (3.10). In applications, one often
assumes that there exist the limits

$\displaystyle{\mathcal{Z}}_{-}^{(-)}$
$\displaystyle\coloneqq\lim_{t\to-\infty}{\mathcal{Z}}_{-}^{(-)}(t),$
$\displaystyle{\mathcal{Z}}_{+}^{(+)}$
$\displaystyle\coloneqq\lim_{t\to+\infty}{\mathcal{Z}}_{+}^{(+)}(t),$

however in what follows this will not be necessary.

######  Definition 3.3.

We say that _asymptotic complementarity_ holds if for some (and thus by (3.10)
for all) $t\in I$,

$\bigl{(}{\mathcal{Z}}_{-}^{(-)}(t),{\mathcal{Z}}_{+}^{(+)}(t)\bigr{)}$

is a pair of complementary subspaces of ${\mathcal{W}}$.

######  Definition 3.4.

Suppose that asymptotic complementarity holds. Then we define

$\displaystyle\Lambda^{(\boxplus-)}(t),$ $\displaystyle\quad\text{the
projection onto ${\mathcal{Z}}_{+}^{(+)}(t)$ along
${\mathcal{Z}}_{-}^{(-)}(t)$},$ (3.11a)
$\displaystyle\Lambda^{(+\boxminus)}(t),$ $\displaystyle\quad\text{the
projection onto ${\mathcal{Z}}_{-}^{(-)}(t)$ along
${\mathcal{Z}}_{+}^{(+)}(t)$},$ (3.11b)

to be the pair of projections associated to the direct sum decomposition
${\mathcal{W}}={\mathcal{Z}}_{+}^{(+)}(t)\oplus{\mathcal{Z}}_{-}^{(-)}(t)$.

By (3.10), we clearly have

$\displaystyle R(t,s)\Lambda^{(\boxplus-)}(s)$
$\displaystyle=\Lambda^{(\boxplus-)}(t)R(t,s),$ $\displaystyle
R(t,s)\Lambda^{(+\boxminus)}(s)$
$\displaystyle=\Lambda^{(+\boxminus)}(t)R(t,s),$

######  Definition 3.5.

If asymptotic complementarity holds we define the operator
$E^{(+-)}:C_{\mathrm{c}}(I,{\mathcal{W}})\to C(I,{\mathcal{W}})$ as in (3.6)
by its temporal integral kernel

$E^{(+-)}(t,s)\coloneqq\theta(t-s)R(t,s)\Lambda^{(\boxplus-)}(s)-\theta(s-t)R(t,s)\Lambda^{(+\boxminus)}(s).$
(3.12)

$E^{(+-)}$ is called the _${\mathcal{Z}}_{+}^{(+)}$ -out
${\mathcal{Z}}_{-}^{(-)}$-in inverse_.

Heuristically, $E^{(+-)}$ is an inverse in the sense of (3.5b). If $I$ is
finite, then $E^{(+-)}$ is bounded on $L^{2}(I,{\mathcal{W}})$.

Asymptotic complementarity is trivially satisfied for the pairs
$({\mathcal{W}},\\{0\\})$ and $(\\{0\\},{\mathcal{W}})$. The corresponding
inverses are simply $E^{\vee}$ and $E^{\wedge}$, defined in Def. 3.2.

In the following, it will be convenient to consider inverses associated to two
pairs of closed subspaces. The construction and the notation for the second
inverse will be consistent with the first one, nevertheless, to avoid
misunderstanding let us make it explicit.

For finite $t_{-},t_{+}$, let
$({\mathcal{Z}}_{+}^{(-)},{\mathcal{Z}}_{-}^{(+)})$ be another pair of closed
subspaces of ${\mathcal{W}}$. Sometimes we will call ${\mathcal{Z}}_{+}^{(-)}$
the out antiparticle space and ${\mathcal{Z}}_{-}^{(+)}$ the in particle
space. Similarly, as above, we define

$\displaystyle{\mathcal{Z}}_{-}^{(+)}(t)$ $\displaystyle\coloneqq
R(t,t_{-}){\mathcal{Z}}_{-}^{(+)},$ (3.13a)
$\displaystyle{\mathcal{Z}}_{+}^{(-)}(t)$ $\displaystyle\coloneqq
R(t,t_{+}){\mathcal{Z}}_{+}^{(-)},$ (3.13b)

so that

$\displaystyle R(t,s){\mathcal{Z}}_{-}^{(+)}(s)$
$\displaystyle={\mathcal{Z}}_{-}^{(+)}(t),$ (3.14a) $\displaystyle
R(t,s){\mathcal{Z}}_{+}^{(-)}(s)$ $\displaystyle={\mathcal{Z}}_{+}^{(-)}(t).$
(3.14b)

If $t_{\pm}=\pm\infty$, we suppose that
$\big{(}{\mathcal{Z}}_{+}^{(-)}(t),{\mathcal{Z}}_{-}^{(+)}(t)\big{)}$ is a
family of pairs of subspaces satisfying (3.14). If asymptotic complementarity
holds for $({\mathcal{Z}}_{+}^{(-)},{\mathcal{Z}}_{-}^{(+)})$, we define the
projections
$\big{(}\Lambda^{(\boxminus+)}(t),\Lambda^{(-\boxplus)}(t)\big{)}$. The
operator $E^{(-+)}:C_{\mathrm{c}}(I,{\mathcal{W}})\to C(I,{\mathcal{W}})$ is
defined as in (3.6) by its temporal integral kernel

$E^{(-+)}(t,s)\coloneqq\theta(t-s)R(t,s)\Lambda^{(\boxminus+)}(s)-\theta(s-t)R(t,s)\Lambda^{(-\boxplus)}(s).$
(3.15)

$E^{(-+)}$ is called the _${\mathcal{Z}}_{+}^{(-)}$ -out
${\mathcal{Z}}_{-}^{(+)}$-in inverse_.

######  Remark 3.6.

Let us explain the notation that we are using.

$+$ and $-$ without parenthesis are related to $t_{+}$ or “out”, resp. $t_{-}$
or “in”. $({+})$ and $({-})$ inside parenthesis denote the “particle space”,
resp. the “antiparticle space”.

The time in the superscript flows from the right to the left. Thus
$\Lambda^{(\boxplus-)}(t)$ corresponds to the antiparticle condition set at
$t_{-}$ and the particle condition set at $t_{+}$. We put $+$ in a box inside
the parenthesis to indicate that this is the projection onto the particles (at
time $t$).

###  3.5 In/out bisolutions associated with projections

For finite $t_{+},t_{-}$, suppose we have two pairs of complementary subspaces

$({\mathcal{Z}}_{-}^{(-)},{\mathcal{Z}}_{-}^{(+)}),\quad({\mathcal{Z}}_{+}^{(-)},{\mathcal{Z}}_{+}^{(+)}).$
(3.16)

Let $(\Pi_{-}^{(-)},\Pi_{-}^{(+)}),$ $(\Pi_{+}^{(-)},\Pi_{+}^{(+)})$ be the
corresponding pairs of coplementary projections and $S_{-},S_{+}$ the
corresponding involutions.

As in (3.9) and (3.13), we set

$\displaystyle{\mathcal{Z}}_{\pm}^{(+)}(t)$ $\displaystyle\coloneqq
R(t,t_{\pm}){\mathcal{Z}}_{\pm}^{(+)},$ (3.17a)
$\displaystyle{\mathcal{Z}}_{\pm}^{(-)}(t)$ $\displaystyle\coloneqq
R(t,t_{\pm}){\mathcal{Z}}_{\pm}^{(-)}.$ (3.17b)

so that

$\displaystyle R(t,s){\mathcal{Z}}_{\pm}^{(+)}(s)$
$\displaystyle={\mathcal{Z}}_{\pm}^{(+)}(t),$ (3.18a) $\displaystyle
R(t,s){\mathcal{Z}}_{\pm}^{(-)}(s)$
$\displaystyle={\mathcal{Z}}_{\pm}^{(-)}(t).$ (3.18b)

If $t_{\pm}=\pm\infty$, we assume that

$\big{(}{\mathcal{Z}}_{-}^{(-)}(t),{\mathcal{Z}}_{-}^{(+)}(t)\big{)},\quad\big{(}{\mathcal{Z}}_{+}^{(-)}(t),{\mathcal{Z}}_{+}^{(+)}(t)\big{)}$
(3.19)

are two pairs of spaces satisfying (3.18), which are complementary for one
(hence for all) $t$.

Let $\big{(}\Pi_{\pm}^{(-)}(t),\Pi_{\pm}^{(+)}(t)\big{)}$ be the pairs of
coplementary projections corresponding to
$\big{(}{\mathcal{Z}}_{-}^{(-)}(t),{\mathcal{Z}}_{-}^{(+)}(t)\big{)}$ and
$S_{\pm}(t)$ the corresponding involutions. Clearly,

$\displaystyle S_{\pm}(t)$ $\displaystyle=R(t,s)S_{\pm}(s)R(s,t),$ (3.20a)
$\displaystyle\Pi_{\pm}^{(+)}(t)$ $\displaystyle\coloneqq
R(t,s)\Pi_{\pm}^{(+)}(s)R(s,t),$ (3.20b) $\displaystyle\Pi_{\pm}^{(-)}(t)$
$\displaystyle\coloneqq R(t,s)\Pi_{\pm}^{(-)}(s)R(s,t).$ (3.20c)

######  Definition 3.7.

We define

$\displaystyle E_{\pm}^{(+)}(t,s)$ $\displaystyle\coloneqq
R(t,s)\Pi_{\pm}^{(+)}(s),$ (3.21a) $\displaystyle E_{\pm}^{(-)}(t,s)$
$\displaystyle\coloneqq-R(t,s)\Pi_{\pm}^{(-)}(s).$ (3.21b)

Define the operators $E_{\pm}^{(+)}$ and
$E_{\pm}^{(-)}:C_{\mathrm{c}}(I,{\mathcal{W}})\to C(I,{\mathcal{W}})$ by their
temporal integral kernels (3.21). $E_{\pm}^{(+)}$ are called the
_$\Pi_{\pm}^{(+)}$ -in/out bisolutions_ and $E_{\pm}^{(-)}$ are called the
_$\Pi_{\pm}^{(-)}$ -in/out bisolutions_.

Clearly, we have

$E^{\mathrm{PJ}}=E_{\pm}^{(+)}-E_{\pm}^{(-)},$ (3.22)

which is analogous to (1.4b).

In applications of pseudo-unitary evolutions to QFT, we usually define
$\Pi_{\pm}^{(+)}$ and $\Pi_{\pm}^{(-)}$ by the spectral projections of the
in/out generator $B(\pm\infty)$ onto the positive/negative halfline, see Def.
4.36. $E^{(+-)}$ will then be called the _Feynman propagator_ and $E^{(-+)}$
the _anti-Feynman propagator_ , see Def. 4.36.

The reason for the minus signs in (3.21b) will become clear in the context of
the pseudo-unitary structure.

###  3.6 Identities involving bisolutions and inverses

We make all assumptions of Subsect. 3.5. In addition we suppose that
asymptotic complementarity holds for both pairs of subspaces. Using the
results of Prop. 2.7, we have the following:

######  Proposition 3.8.

The projections defined in (3.11) are given explicitly by

$\displaystyle\Lambda^{(\boxplus-)}(t)$
$\displaystyle=\Pi_{+}^{(+)}(t)\Upsilon(t)^{-1}\Pi_{-}^{(+)}(t),\quad$
$\displaystyle\Lambda^{(+\boxminus)}(t)$
$\displaystyle=\Pi_{-}^{(-)}(t)\Upsilon(t)^{-1}\Pi_{+}^{(-)}(t),$
$\displaystyle\text{and}\quad\Lambda^{(-\boxplus)}(t)$
$\displaystyle=\Pi_{-}^{(+)}(t)\Upsilon(t)^{-1}\Pi_{+}^{(+)}(t),\quad$
$\displaystyle\Lambda^{(\boxminus+)}(t)$
$\displaystyle=\Pi_{+}^{(-)}(t)\Upsilon(t)^{-1}\Pi_{-}^{(-)}(t),$

where $\Upsilon(t)$ is the invertible operator defined by

$\displaystyle\Upsilon(t)$
$\displaystyle\coloneqq\frac{1}{4}\bigl{(}2+S_{-}(t)S_{+}(t)+S_{+}(t)S_{-}(t)\bigr{)}.$

Observe that $\Upsilon$ has the properties

$\displaystyle\Upsilon(t)$ $\displaystyle=R(t,s)\Upsilon(s)R(s,t),$
$\displaystyle\Upsilon(t)S_{\pm}(t)$ $\displaystyle=S_{\pm}(t)\Upsilon(t).$

Thus we have defined the inverses $E^{\vee}$, $E^{\wedge}$, $E^{(+-)}$,
$E^{(-+)}$ and the bisolutions $E^{\mathrm{PJ}}$, $E_{-}^{(+)}$,
$E_{-}^{(-)}$, $E_{+}^{(+)}$, $E_{+}^{(-)}$. They satisfy the relations (3.8)
and (3.22). As described in the introduction, in the setting of the Minkowski
space, and more generally, of a stationary spacetime, they satisfy several
other identities. These identities do not hold in general. Instead, we have
the following relations:

######  Proposition 3.9.

$(E_{+}^{(+)}+E_{+}^{(-)}-E_{-}^{(+)}-E_{-}^{(-)})(t,s)=R(t,s)(S_{+}-S_{-})(s),$
(3.23)
$\begin{split}&(E^{(+-)}-E^{(-+)})(t,s)-\frac{1}{2}(E_{+}^{(+)}+E_{+}^{(-)}+E_{-}^{(+)}+E_{-}^{(-)})(t,s)\\\
&=\frac{1}{8}R(t,s)\Upsilon(s)^{-1}\big{[}S_{+}(s)-S_{-}(s),[S_{+}(s),S_{-}(s)]\big{]}.\end{split}$
(3.24)
$(E^{(+-)}+E^{(-+)}-E^{\vee}-E^{\wedge})(t,s)=\frac{1}{4}R(t,s)\Upsilon(s)^{-1}[S_{-}(s),S_{+}(s)].$
(3.25)

###### Proof.

(3.23) is straightforward.

Obviously, the difference of two inverses is a bisolution. The temporal
kernels of the following bisolutions have very simple forms:

$\displaystyle(E^{(+-)}-E^{\wedge})(t,s)$
$\displaystyle=R(t,s)\Lambda^{(\boxplus-)}(s),$ (3.26)
$\displaystyle(E^{(+-)}-E^{\vee})(t,s)$
$\displaystyle=-R(t,s)\Lambda^{(+\boxminus)}(s),$ (3.27)
$\displaystyle(E^{(-+)}-E^{\wedge})(t,s)$
$\displaystyle=R(t,s)\Lambda^{(\boxminus+)}(s),$ (3.28)
$\displaystyle(E^{(-+)}-E^{\vee})(t,s)$
$\displaystyle=-R(t,s)\Lambda^{(-\boxplus)}(s).$ (3.29)

Taking the differences of (3.26) and (3.28), and of (3.27) and (3.29) we
obtain the following identities:

$\displaystyle E^{(+-)}(t,s)-E^{(-+)}(t,s)$
$\displaystyle=R(t,s)(\Lambda^{(\boxplus-)}-\Lambda^{(\boxminus+)})(s)=R(t,s)\Upsilon(s)^{-1}S_{+}(s)$
$\displaystyle=R(t,s)(\Lambda^{(-\boxplus)}-\Lambda^{(+\boxminus)})(s)=R(t,s)\Upsilon(s)^{-1}S_{-}(s)$
$\displaystyle=\frac{1}{2}R(t,s)\Upsilon(s)^{-1}(S_{+}+S_{-})(s)$
$\displaystyle=R(t,s)\Big{(}\frac{1}{2}(S_{+}+S_{-})(s)+\frac{1}{8}\Upsilon(s)^{-1}(S_{+}+S_{-}-S_{-}S_{+}S_{-}-S_{+}S_{-}S_{+})(s)\Big{)},$

This yields (LABEL:rwq4a).

Taking the sum of (3.26) and (3.29) we obtain

$\displaystyle(E^{(+-)}+E^{(-+)}-E^{\vee}-E^{\wedge})(t,s)$ $\displaystyle=$
$\displaystyle R(t,s)(\Lambda^{(\boxplus-)}-\Lambda^{(-\boxplus)})(s)$
$\displaystyle=$ $\displaystyle
R(t,s)\Upsilon(s)^{-1}(\Pi_{+}^{(+)}\Pi_{-}^{(+)}-\Pi_{-}^{(+)}\Pi_{+}^{(+)})(s),$

which yields (3.25). ∎

Note that for the standard choice of propagators in a stationary QFT the right
hand sides of (3.23), (LABEL:rwq4a) and (3.25) vanish. It is easy to see that
they do not have to vanish in general.

The identities (3.25) and (LABEL:rwq4a) simplify in some important situations:

######  Proposition 3.10.

Assume that asymptotic complementarity holds. Further, suppose that for any
(and hence for all) $t\in I$

$S_{-}(t)S_{+}(t)=S_{+}(t)S_{-}(t).$ (3.30)

Then

$\displaystyle E^{(+-)}+E^{(-+)}$ $\displaystyle=E^{\vee}+E^{\wedge},$ (3.31a)
$\displaystyle E^{(+-)}-E^{(-+)}$
$\displaystyle=\frac{1}{2}(E_{+}^{(+)}+E_{+}^{(-)}+E_{-}^{(+)}+E_{-}^{(-)}).$
(3.31b)

Eq. (3.30) is satisfied in a number of interesting situations. In particular,
if the evolution is autonomous, it is natural to assume that
$S_{+}=S_{-}=:S_{\bullet}$, requiring that it commutes with the generator $B$.
Then $E_{\pm}^{(+)}$ and $E_{\pm}^{(-)}$ collapse to two bisolutions:

$\displaystyle E_{+}^{(+)}$ $\displaystyle=E_{-}^{(+)}\eqqcolon E^{(+)},$
$\displaystyle E_{+}^{(-)}$ $\displaystyle=E_{-}^{(-)}\eqqcolon E^{(-)}.$

Thus, in the autonomous case, the identity (3.31a) holds and (3.31b) can be
rewritten as

$\displaystyle E^{(+-)}-E^{(-+)}$ $\displaystyle=E^{(+)}+E^{(-)}.$

###  3.7 Almost unitary evolutions on Hilbertizable spaces

So far we discussed generators of an evolution only in a heuristic way. In
this subsection we will describe a setting that allows us to make this concept
precise. In view of our applications, we will introduce generators of
evolutions on Hilbertizable spaces that one might call _almost unitary
evolutions_.

For the remainder of this section, we suppose that ${\mathcal{W}}$ and
${\mathcal{V}}$ are Hilbertizable spaces such that ${\mathcal{V}}$ is densely
and continuously embedded in ${\mathcal{W}}$. As described in Subsect. 2.4, by
interpolation, we can introduce the scale of Hilbertizable spaces
${\mathcal{W}}_{\alpha}$, $\alpha\in[0,1]$ with
${\mathcal{W}}={\mathcal{W}}_{0}$ and ${\mathcal{V}}={\mathcal{W}}_{1}$.

######  Theorem 3.11 (cf. Thm. C.10 of [7]).

Let $\\{B(t)\\}_{t\in I}$ be a family of densely defined, closed operators on
${\mathcal{W}}$. Suppose that the following conditions are satisfied:

  1. (a)

${\mathcal{V}}\subset\operatorname{{\mathcal{D}}}\big{(}B(t)\big{)}$ so that
$B(t)\in B({\mathcal{V}},{\mathcal{W}})$ and $I\ni t\mapsto B(t)\in
B({\mathcal{V}},{\mathcal{W}})$ is norm-continuous.

  2. (b)

For every $t\in I$, scalar products $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{{\mathcal{W}},t}$ and $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{{\mathcal{V}},t}$ compatible with ${\mathcal{W}}$
resp. ${\mathcal{V}}$ have been chosen. Denote the corresponding Hilbert
spaces ${\mathcal{W}}_{t}$ and ${\mathcal{V}}_{t}$.

  3. (c)

$B(t)$ is self-adjoint in the sense of ${\mathcal{W}}_{t}$ and the part
$\tilde{B}(t)$ of $B(t)$ in ${\mathcal{V}}$ is self-adjoint in the sense of
${\mathcal{V}}_{t}$.

  4. (d)

For a positive $C\in L_{\mathrm{loc}}^{1}(I)$ and all $s,t\in I$,

$\displaystyle\lVert w\rVert_{{\mathcal{W}},t}$ $\displaystyle\leq\lVert
w\rVert_{{\mathcal{W}},s}\exp\left\lvert\int_{s}^{t}C(r)\mathop{}\\!\mathrm{d}r\right\rvert,$
$\displaystyle\lVert v\rVert_{{\mathcal{V}},t}$ $\displaystyle\leq\lVert
v\rVert_{{\mathcal{V}},s}\exp\left\lvert\int_{s}^{t}C(r)\mathop{}\\!\mathrm{d}r\right\rvert.$

Then there exists a unique family of bounded operators $\\{R(t,s)\\}_{s,t\in
I}$, on ${\mathcal{W}}$ with the following properties:

  1. (i)

For all $r,s,t\in I$, we have the identities (3.2).

  2. (ii)

$R(t,s)$ is ${\mathcal{W}}$-strongly continuous. It preserves ${\mathcal{V}}$
and is ${\mathcal{V}}$-strongly continuous. Hence it preserves
${\mathcal{W}}_{\alpha}$, $0\leq\alpha\leq 1$. Moreover,

$\lVert
R(t,s)\rVert_{\alpha,t}\leq\exp\left\lvert\int_{s}^{t}2C(r)\mathop{}\\!\mathrm{d}r\right\rvert,\quad
s,t\in I.$

  3. (iii)

For all $w\in{\mathcal{V}}$ and $s,t\in I$,

$\displaystyle\mathrm{i}\partial_{t}R(t,s)w$ $\displaystyle=B(t)R(t,s)w,$
(3.32a) $\displaystyle-\mathrm{i}\partial_{s}R(t,s)w$
$\displaystyle=R(t,s)B(s)w,$ (3.32b)

where the derivatives are in the strong topology of ${\mathcal{W}}$.

We call $\\{R(t,s)\\}_{s,t\in I}$ the _evolution_ generated by $B(t)$.

Note that, if $t_{\pm}$ are finite, the above theorem remains true if we
everywhere replace $I$ with $I^{\mathrm{cl}}=[t_{-},t_{+}]$, and
$L_{\mathrm{loc}}^{1}]t_{-},t_{+}[$ with $L^{1}[t_{-},t_{+}]$, provided that
we consider only the right/left-sided derivatives at $t_{-}/t_{+}$.

Sometimes it is convenient to use an easy generalization of Thm 3.11, where we
perturbed the generator by a bounded perturbation. It is also proven in [7].

######  Theorem 3.12.

Suppose that $\\{B_{0}(t)\\}_{t\in I}$ satisfies all the assumptions of Thm
3.11 and $I\ni t\mapsto V(t)\in B({\mathcal{W}})$ is a norm continuous family
operators. Let $B(t):=B_{0}(t)+V(t)$. Then there exists a unique family of
bounded operators $\\{R(t,s)\\}_{s,t\in I}$, on ${\mathcal{W}}$ satisfying all
properties of Thm 3.11.

If we want to pass to the real case, we use the following obvious fact:

######  Proposition 3.13.

If ${\mathcal{W}}$ has a conjugation which preserves ${\mathcal{V}}$, then
$R(t,s)$ is real for $t,s\in I$ if and only if its generator $B(t)$ is anti-
real for all $t\in I$.

###  3.8 Rigorous concept of a bisolution and inverse

Under the assumptions of Thm. 3.11 it is possible to propose a rigorous
version of a concept of a (left) bisolution and a (left) inverse, and check
that they are satisfied by the $E^{\bullet}$ that we have constructed.

######  Proposition 3.14.

  1. (1)

Let $v\in C_{\mathrm{c}}(I,{\mathcal{V}})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{W}})$. Then for
$E^{\bullet}=E^{\mathrm{PJ}},E_{-}^{(+)},E_{-}^{(-)}$,
$E_{+}^{(+)},E_{+}^{(-)}$ we have

$\displaystyle E^{\bullet}\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}v$
$\displaystyle=0,$ (3.33a) and for
$E^{\bullet}=E^{\wedge},E^{\vee},E^{(+-)},E^{(-+)}$ we have
$\displaystyle\quad E^{\bullet}\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}v$
$\displaystyle=v.$ (3.33b)

  2. (2)

Let $w\in C_{\mathrm{c}}(I,{\mathcal{V}})$. Then for
$E^{\bullet}=E^{\mathrm{PJ}}$ we have

$\displaystyle\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}E^{\bullet}w$
$\displaystyle=0,$ (3.34a) and for $E^{\bullet}=E^{\wedge},E^{\vee}$ we have
$\displaystyle\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}E^{\bullet}w$
$\displaystyle=w.$ (3.34b)

###### Proof.

Let us prove (1) for $E^{(+-)}$:

$\displaystyle\Big{(}E^{(+-)}\big{(}\partial_{s}+\mathrm{i}B(s)\big{)}v\Big{)}(t)$
$\displaystyle=$
$\displaystyle\int_{t_{-}}^{t}\Lambda^{(\boxplus-)}(t)R(t,s)\big{(}\partial_{s}+\mathrm{i}B(s)\big{)}v(s)\mathop{}\\!\mathrm{d}s$
$\displaystyle-\int_{t}^{t_{+}}\Lambda^{(+\boxminus)}(t)R(t,s)\big{(}\partial_{s}+\mathrm{i}B(s)\big{)}v(s)\mathop{}\\!\mathrm{d}s$
$\displaystyle=$
$\displaystyle\int_{t_{-}}^{t}\Lambda^{(\boxplus-)}(t)\partial_{s}\big{(}R(t,s)v(s)\big{)}\mathop{}\\!\mathrm{d}s$
$\displaystyle-\int_{t}^{t_{+}}\Lambda^{(+\boxminus)}(t)\partial_{s}\big{(}R(t,s)v(s)\big{)}\mathop{}\\!\mathrm{d}s$
$\displaystyle=$
$\displaystyle\Lambda^{(\boxplus-)}(t)v(t)+\Lambda^{(+\boxminus)}(t)v(t)\quad=\quad
v(t).$

Let us prove (2) for $E^{\vee}$:

$\displaystyle\Big{(}\big{(}\partial_{t}+\mathrm{i}B(t)\big{)}E^{\vee}w\Big{)}(t)$
$\displaystyle=$
$\displaystyle\big{(}\partial_{t}+\mathrm{i}B(t)\big{)}\int_{t_{-}}^{t}R(t,s)w(s)\mathop{}\\!\mathrm{d}s$
$\displaystyle=$ $\displaystyle
R(t,t)w(t)+\int_{t_{-}}^{t}\big{(}\partial_{t}+\mathrm{i}B(t)\big{)}R(t,s)w(s)\mathop{}\\!\mathrm{d}s\quad=\quad
w(t).$

∎

##  4 Evolutions on pseudo-unitary spaces

Pre-pseudo-unitary spaces are Hilbertizable spaces with a distinguished
bounded Hermitian form. They can be viewed as complexifications of pre-
symplectic spaces – real spaces with a distinguished bounded antisymmetric
form.

In practice, one usually assumes that the Hermitian or pre-symplectic form is
non-degenerate. Then these spaces are called pseudo-unitary, resp. symplectic.
A transformation preserving the structure of a pseudo-unitary, resp.
symplectic space is called pseudo-unitary, resp. symplectic.

Krein spaces constitute an especially well-behaved class of pseudo-unitary
spaces. The main topic of this section are pseudo-unitary transformations on
Krein spaces. We will see that such transformations enjoy special properties.

We will also discuss generators of 1-parameter groups preserving the pseudo-
unitary structure, called pseudo-unitary generators. In particular, we will
introduce the so-called stable pseudo-unitary generators, which possess
positive Hamiltonians. They are distinguished both on physical and
mathematical grounds. Especially good properties have strongly stable pseudo-
unitary generators, whose positive Hamiltonians are bounded away from zero.

The Krein structure adds interesting new features to bisolutions and inverses
of $M:=\partial_{t}+\mathrm{i}B(t)$. The most interesting new fact is the
automatic validity of asymptotic complementarity if the “in particle space” is
m-positive and the “out antiparticle space” is m-negative. This implies the
existence of the Feynman and anti-Feynman inverses.

In Subsection 4.7 we give a basic criterion for a generator to define pseudo-
unitary evolution (or, as is stated there, to preserve a given Hermitian form,
which is slightly more general). This is then used in Subsection 4.9 to
describe closed realizations of the operator $M$ if the interval $I$ is
finite.

In practice Krein spaces are often constructed by the interpolation from a
nested pair of Hilbertizable spaces equipped with a Hermitian pairing. This
approach allows us to construct a pseudo-unitary dynamics under relatively
weak conditions. We describe this approach in Subsects 4.10 and 4.11.

###  4.1 Symplectic and pseudo-unitary spaces

######  Definition 4.1.

A _pre-symplectic space_ is a real vector space ${\mathcal{Y}}$ equipped with
an antisymmetric form $\omega$, called a _pre-symplectic form_

${\mathcal{Y}}\times{\mathcal{Y}}\ni(v,w)\mapsto\langle v\mkern 2.0mu|\mkern
2.0mu\mathopen{}\omega w\rangle\in{{\mathbb{R}}}.$

If $\omega$ is non-degenerate, then a pre-symplectic space is called
_symplectic_. If the dimension of ${\mathcal{Y}}$ is infinite, we assume that
${\mathcal{Y}}$ is Hilbertizable and $\omega$ is bounded.

######  Definition 4.2.

We will say that a bounded invertible operator $R$ on a pre-symplectic space
$({\mathcal{Y}},\omega)$ _preserves $\omega$_ if

$\langle Rv\mkern 2.0mu|\mkern 2.0mu\mathopen{}\omega Rw\rangle=\langle
v\mkern 2.0mu|\mkern 2.0mu\mathopen{}\omega w\rangle.$

If in addition $\omega$ is non-degenerate, we will say that $R$ is
_symplectic_.

######  Definition 4.3.

A _pre-pseudo-unitary space_ is a complex vector space ${\mathcal{W}}$
equipped with a Hermitian form $Q$

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

If $Q$ is non-degenerate, then a pre-pseudo-unitary space is called _pseudo-
unitary_. If the dimension of ${\mathcal{W}}$ is infinite, we assume that
${\mathcal{W}}$ is Hilbertizable and $Q$ is bounded.

######  Definition 4.4.

We will say that a bounded invertible operator $R$ on $({\mathcal{W}},Q)$
_preserves $Q$_ if

$(Rv\mkern 2.0mu|\mkern 2.0mu\mathopen{}QRw)=(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw).$

If in addition $Q$ is non-degenerate, we will say that $R$ is _pseudo-
unitary_.

Note that even if one starts from a real (pre-)symplectic space, it is useful
to consider its complexification. In Subsect. 2.6 we described how to pass
from the real to complex formalism. In this section we will treat the complex
formalism as the standard one.

In the context of a pre-pseudo-unitary space treated as a complexification of
a pre-symplectic space it is natural to consider conjugations that anti-
preserve $Q$, that is $\overline{(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw)}=-(\overline{v}\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q\overline{w}),$ see (2.14).

######  Definition 4.5.

An antilinear involution $v\mapsto\overline{v}$ on a pseudo-unitary space
$({\mathcal{W}},Q)$ which anti-preserves $Q$ and such that there exists a
compatible scalar product $(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)_{\bullet}$ satisfying

$(\overline{v}\mkern 2.0mu|\mkern
2.0mu\mathopen{}\overline{w})_{\bullet}=\overline{(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}w)}_{\bullet},$

will be called a _conjugation on $({\mathcal{W}},Q)$_.

By (2.9), given a conjugation we can define the real subspace
${\mathcal{W}}_{{\mathbb{R}}}$ of ${\mathcal{W}}$. The restriction of $Q$ to
${\mathcal{W}}_{{\mathbb{R}}}$ is clearly a pre-symplectic space.

###  4.2 Admissible involutions and Krein spaces

Let $({\mathcal{W}},Q)$ be a pre-pseudo-unitary space.

######  Definition 4.6.

A (bounded) involution $S_{\bullet}$ on ${\mathcal{W}}$ will be called
_admissible_ if it preserves $Q$ and the scalar product

${(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}w)}_{\bullet}\coloneqq(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}QS_{\bullet}w)=(S_{\bullet}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw)$ (4.1)

is compatible with the Hilbertizable structure of ${\mathcal{W}}$. Sometimes
we will write ${\mathcal{W}}_{\bullet}$ to denote the space ${\mathcal{W}}$
equipped with the scalar product (4.1).

######  Definition 4.7.

A pre-pseudo-unitary space is called a _Krein space_ if it possesses an
admissible involution.

Clearly, every Krein space is pseudo-unitary.

######  Proposition 4.8.

If $S_{\bullet}$ is an admissible involution on $({\mathcal{W}},Q)$, then
$S_{\bullet}$ is self-adjoint and unitary on ${\mathcal{W}}_{\bullet}$.

For any admissible involution $S_{\bullet}$, we define the corresponding
_particle projection_ $\Pi_{\bullet}^{(+)}$ and _particle space_
${\mathcal{Z}}_{\bullet}^{(+)}$, as well as the _antiparticle projection_
$\Pi_{\bullet}^{(-)}$ and _antiparticle space_
${\mathcal{Z}}_{\bullet}^{(-)}$, as in (2.16). Note the following relations:

$\displaystyle{(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}w)}_{\bullet}$
$\displaystyle={(\Pi_{\bullet}^{(+)}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}\Pi_{\bullet}^{(+)}w)}_{\bullet}+{(\Pi_{\bullet}^{(-)}v\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Pi_{\bullet}^{(-)}w)}_{\bullet},$
$\displaystyle(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)$
$\displaystyle={(\Pi_{\bullet}^{(+)}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}\Pi_{\bullet}^{(+)}w)}_{\bullet}-{(\Pi_{\bullet}^{(-)}v\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Pi_{\bullet}^{(-)}w)}_{\bullet}.$

Let us make some additional comments on the Krein spaces with conjugations.

######  Proposition 4.9.

Suppose that $({\mathcal{W}},Q)$ is a Krein space with a conjugation. If
$S_{\bullet}$ is an admissible anti-real involution, then
$\mathrm{i}S_{\bullet}$ is real and we have

$\overline{\Pi_{\bullet}^{(+)}}=\Pi_{\bullet}^{(-)},\quad\overline{{\mathcal{Z}}_{\bullet}^{(+)}}={\mathcal{Z}}_{\bullet}^{(-)},$

so that
${\mathcal{W}}={\mathcal{Z}}_{\bullet}^{(+)}\oplus\overline{{\mathcal{Z}}_{\bullet}^{(+)}}$.

###  4.3 Basic constructions in Krein spaces

Let $({\mathcal{W}},Q)$ be a Krein space.

######  Definition 4.10.

Let ${\mathcal{Z}}\subset{\mathcal{W}}$. We define its $Q$-orthogonal
complement as follows:

${\mathcal{Z}}^{\perp Q}:=\\{w\in{\mathcal{W}}\mid(w|Qv)=0,\quad
v\in{\mathcal{Z}}\\}.$

If $(\cdot|\cdot)_{\bullet}$ is a scalar product, we also have the
$\bullet$-orthogonal complement

${\mathcal{Z}}^{\perp\bullet}:=\\{w\in{\mathcal{W}}\mid(w|v)_{\bullet}=0,\quad
v\in{\mathcal{Z}}\\}.$

######  Proposition 4.11.

  1. (1)

If ${\mathcal{Z}}$ is a closed subspace, then so is ${\mathcal{Z}}^{\perp Q}$
and $({\mathcal{Z}}^{\perp Q})^{\perp Q}={\mathcal{Z}}$.

  2. (2)

If ${\mathcal{Z}}_{1},{\mathcal{Z}}_{2}$ are complementary in ${\mathcal{W}}$,
then so are ${\mathcal{Z}}_{1}^{\perp Q},{\mathcal{Z}}_{2}^{\perp Q}$.

  3. (3)

If $(v|Qw)=(v|S_{\bullet}w)_{\bullet}$, then ${\mathcal{Z}}^{\perp
Q}=S_{\bullet}{\mathcal{Z}}^{\perp\bullet}$.

######  Definition 4.12.

Let $A\in B({\mathcal{W}})$. We define its $Q$-adjoint as follows:

$(A^{*Q}v|Qw)=(v|QAw),\quad v,w\in{\mathcal{W}}.$

If $(\cdot|\cdot)_{\bullet}$ is a scalar product, we also have the
$\bullet$-adjoint of $A$

$(A^{*\bullet}v|w)_{\bullet}=(v|Aw)_{\bullet},\quad v,w\in{\mathcal{W}}.$

######  Proposition 4.13.

  1. (1)

If $(v|Qw)=(v|S_{\bullet}w)_{\bullet}$, then
$A^{*Q}=S_{\bullet}A^{*\bullet}S_{\bullet}$.

  2. (2)

Let $(\Pi^{(+)},\Pi^{(-)})$ be pair of complementary projections. Then
$(\Pi^{(+)*Q},\Pi^{(-)*Q})$ is also a pair of complementary projections and

$\displaystyle{\mathcal{R}}(\Pi^{(\pm)*Q})={\mathcal{N}}(\Pi^{(\mp)*Q})=$
$\displaystyle{\mathcal{R}}(\Pi^{(\mp)})^{\perp
Q}={\mathcal{N}}(\Pi^{(\pm)})^{\perp Q}.$

######  Proposition 4.14.

Let ${\mathcal{Z}}_{\bullet}^{(+)}$ be a closed subspace of ${\mathcal{W}}$.
Set ${\mathcal{Z}}_{\bullet}^{(-)}:={\mathcal{Z}}_{\bullet}^{(+)\perp Q}$. The
following conditions are equivalent:

  1. (1)

${\mathcal{Z}}_{\bullet}^{(+)}$ satisfies

$\displaystyle v\in{\mathcal{Z}}_{\bullet}^{(+)}$
$\displaystyle\Rightarrow(v|Qv)\geq 0,$ (4.2) $\displaystyle\text{and }\quad
v\in{\mathcal{Z}}_{\bullet}^{(-)}$ $\displaystyle\Rightarrow(v|Qv)\leq 0.$
(4.3)

  2. (2)

${\mathcal{Z}}_{\bullet}^{(+)}$ is a maximal closed subspace of
${\mathcal{W}}$ with the property (4.2).

  3. (3)

The spaces ${\mathcal{Z}}_{\bullet}^{(+)}$ and ${\mathcal{Z}}_{\bullet}^{(-)}$
are complementary, and if $(\Pi_{\bullet}^{(+)},\Pi_{\bullet}^{(-)})$ is the
corresponding pair of projections, then
$S_{\bullet}:=\Pi_{\bullet}^{(+)}-\Pi_{\bullet}^{(-)}$ is an admissible
involution.

######  Definition 4.15.

If ${\mathcal{Z}}_{\bullet}^{(+)}$ satisfies the conditions of Proposition
4.14, then it is called a m-positive subspace. Analogously we define
m-negative subspaces. (The letter “m” is an abbreviation of “maximal”).

###  4.4 Pairs of admissible involutions

Let $S_{1}$, $S_{2}$ be a pair of admissible involutions on a Krein space
$({\mathcal{W}},Q)$. We will describe some structural properties of such a
pair.

Let
$\Pi_{i}^{(+)},\Pi_{i}^{(-)},{\mathcal{Z}}_{i}^{(+)},{\mathcal{Z}}_{i}^{(-)}$,
$i=1,2$, be defined as in (2.16). Set

$K(2,1)\coloneqq S_{2}S_{1},\quad
c(2,1)\coloneqq\Pi_{1}^{(+)}\frac{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-K(2,1)}{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K(2,1)}\Pi_{1}^{(-)},$ (4.4)

where $c(2,1)$ is interpreted as an operator from ${\mathcal{Z}}_{1}^{(-)}$ to
${\mathcal{Z}}_{1}^{(+)}$. For brevity, we will usually write $K$, $c$ for
$K(2,1)$, $c(2,1)$.

######  Proposition 4.16.

$K$ is pseudo-unitary. It is positive with respect to ${(\cdot\mkern
2.0mu|\mkern 2.0mu\mathopen{}\cdot)}_{1}$ and ${(\cdot\mkern 2.0mu|\mkern
2.0mu\mathopen{}\cdot)}_{2}$. We have $\lVert c\rVert<1$ and

$\displaystyle S_{1}KS_{1}=S_{2}KS_{2}$ $\displaystyle=K^{-1},$ (4.5)
$\displaystyle S_{1}\frac{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-K}{{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K}S_{1}=S_{2}\frac{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-K}{{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K}S_{2}$ $\displaystyle=-\frac{{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-K}{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}+K}.$ (4.6)
$\displaystyle\frac{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-K}{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}+K}$
$\displaystyle=\begin{bmatrix}0&c\\\ c^{*}&0\end{bmatrix},$ (4.7a)
$\displaystyle K$ $\displaystyle=\mathrlap{\begin{bmatrix}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+cc^{*})({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-1}&-2c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-c^{*}c)^{-1}\\\
-2c^{*}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-cc^{*})^{-1}&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+c^{*}c)({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-1}\end{bmatrix},}$ (4.7b)
$\displaystyle\Pi_{1}^{(+)}$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ 0&0\end{bmatrix},$ $\displaystyle\quad\Pi_{2}^{(+)}$
$\displaystyle=\begin{bmatrix}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-1}&c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-c^{*}c)^{-1}\\\
-c^{*}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-cc^{*})^{-1}&-c^{*}c({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-1}\end{bmatrix},$ (4.7c)
$\displaystyle\Pi_{1}^{(-)}$ $\displaystyle=\begin{bmatrix}0&0\\\
0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix},$
$\displaystyle\quad\Pi_{2}^{(-)}$
$\displaystyle=\begin{bmatrix}-cc^{*}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-1}&-c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-c^{*}c)^{-1}\\\
c^{*}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-cc^{*})^{-1}&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-1}\end{bmatrix},$ (4.7d) $\displaystyle S_{1}$
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
0&-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix},$ $\displaystyle\quad
S_{2}$ $\displaystyle=\begin{bmatrix}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+cc^{*})({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-1}&2c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-c^{*}c)^{-1}\\\
-2c^{*}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-cc^{*})^{-1}&-({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+c^{*}c)({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-1}\end{bmatrix},$ (4.7e)

where we used the decomposition
${\mathcal{W}}={\mathcal{Z}}_{1}^{(+)}\oplus{\mathcal{Z}}_{1}^{(-)}$.

###### Proof.

$K$ is pseudo-unitary as the product of two pseudo-unitary transformations.
The inequalities

$\displaystyle{(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Kv)}_{1}$
$\displaystyle=(S_{1}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}QS_{2}S_{1}v)={(S_{1}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}S_{1}v)}_{2}\geq a{(S_{1}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}S_{1}v)}_{1}=a{(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}v)}_{1}$
$\displaystyle{(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Kv)}_{2}$
$\displaystyle={(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}v)}_{2}\geq b{(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}v)}_{2}$

with $a,b>0$ show the positivity of $K$. The identity

$S_{1}KS_{1}=S_{1}S_{2}=K^{-1}$

shows (4.5), which by

$S_{1}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-K)({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K)^{-1}S_{1}=({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-K^{-1})({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K^{-1})^{-1}=-({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-K)({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K)^{-1}.$

implies (4.6), from which (4.7a) follows.

From the definition of $c$ (or (4.7a)) we obtain

$\displaystyle K$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&-c\\\ -c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&c\\\ c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}^{-1}$
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&-c\\\
-c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-1}&-c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-c^{*}c)^{-1}\\\
-c^{*}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-cc^{*})^{-1}&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-1}\end{bmatrix}.$

This yields (4.7b).

From $S_{2}=KS_{1}$ we obtain (4.7c), (4.7d) and (4.7e). ∎

It is not difficult to prove that a converse to the second part of the
proposition holds: if $S_{1}$ is an admissible involution and $\lVert
c\rVert<1$, then $S_{2}$ given as in (4.7e) is an admissible involution.

The involutions $S_{1}$ and $S_{2}$ correspond to the pairs of complementary
subspaces $({\mathcal{Z}}_{1}^{(+)},{\mathcal{Z}}_{1}^{(-)})$, resp.
$({\mathcal{Z}}_{2}^{(+)},{\mathcal{Z}}_{2}^{(-)})$. The following proposition
implies the existence of two other direct sum decompositions. This fact plays
an important role in the construction of the (in-out) Feynman inverse.

######  Proposition 4.17.

The pairs of subspaces $({\mathcal{Z}}_{1}^{(+)},{\mathcal{Z}}_{2}^{(-)})$ and
$({\mathcal{Z}}_{2}^{(+)},{\mathcal{Z}}_{1}^{(-)})$ are complementary. Here
are the corresponding projections:

$\displaystyle\Lambda_{12}^{(+)}$
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&c\\\
0&0\end{bmatrix}$ $\displaystyle=\Pi_{1}^{(+)}\Upsilon^{-1}\Pi_{2}^{(+)}$
$\displaystyle\text{ projects onto ${\mathcal{Z}}_{1}^{(+)}$ along
${\mathcal{Z}}_{2}^{(-)}$},$ (4.8a) $\displaystyle\Lambda_{12}^{(-)}$
$\displaystyle=\begin{bmatrix}0&-c\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}$
$\displaystyle=\Pi_{2}^{(-)}\Upsilon^{-1}\Pi_{1}^{(-)}$ $\displaystyle\text{
projects onto ${\mathcal{Z}}_{2}^{(-)}$ along ${\mathcal{Z}}_{1}^{(+)}$},$
(4.8b) $\displaystyle\Lambda_{21}^{(+)}$
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
-c^{*}&0\end{bmatrix}$ $\displaystyle=\Pi_{2}^{(+)}\Upsilon^{-1}\Pi_{1}^{(+)}$
$\displaystyle\text{ projects onto ${\mathcal{Z}}_{2}^{(+)}$ along
${\mathcal{Z}}_{1}^{(-)}$},$ (4.8c) $\displaystyle\Lambda_{21}^{(-)}$
$\displaystyle=\begin{bmatrix}0&0\\\ c^{*}&{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}$
$\displaystyle=\Pi_{1}^{(-)}\Upsilon^{-1}\Pi_{2}^{(-)}$ $\displaystyle\text{
projects onto ${\mathcal{Z}}_{1}^{(-)}$ along ${\mathcal{Z}}_{2}^{(+)}$},$
(4.8d)

where

$\Upsilon^{-1}=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-cc^{*}&0\\\
0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c\end{bmatrix}=\frac{4}{(2+S_{2}S_{1}+S_{1}S_{2})}=\frac{4}{({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K)({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}+K^{-1})}.$ (4.9)

###### Proof.

We apply Prop. 2.6. ∎

We can reformulate Prop. 4.17 as follows.

######  Proposition 4.18.

Let ${\mathcal{Z}}_{1}$ be an m-positive subspace and ${\mathcal{Z}}_{2}$ an
m-negative space. Then they are complementary.

###### Proof.

By Prop. 4.14 there exist admissible involutions $S_{1}$ and $S_{2}$ such that
${\mathcal{Z}}_{1}={\mathcal{Z}}_{1}^{(+)}$ and
${\mathcal{Z}}_{2}={\mathcal{Z}}_{2}^{(-)}$. Hence, it suffices to apply Prop.
4.17. ∎

Remember that $K$ is positive (with respect to both ${\mathcal{W}}_{1}$ and
${\mathcal{W}}_{2}$.) Hence it possesses a unique positive square root. We set

$M(2,1)\coloneqq\sqrt{K(2,1)}.$

For brevity, we will write $M$ for $M(2,1)$. $M$ is a natural similarity
transformation between $S_{1}$ and $S_{2}$ (see e.g. [5]):

######  Proposition 4.19.

We have

$\displaystyle S_{2}$ $\displaystyle=MS_{1}M^{-1},$ (4.10) $\displaystyle M$
$\displaystyle=\begin{bmatrix}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-\frac{1}{2}}&-c({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\\\ -c^{*}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-\frac{1}{2}}&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\end{bmatrix}$ (4.11)
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&-c\\\
0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{2}}&0\\\ 0&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ -c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(4.12)

###### Proof.

By definition of $K$, it holds that

$KS_{1}K=S_{1}.$ (4.13)

Since $K$ is invertible, (4.13) can be rewritten as

$K=S_{1}K^{-1}S_{1}.$

The positive square root of a positive operator is a unitary invariant, and
$S_{1}$ is a self-adjoint unitary. Therefore,

$M=S_{1}M^{-1}S_{1}.$

Using this, we obtain

$S_{2}=KS_{1}=M^{2}S_{1}=MS_{1}M^{-1}.$

We easily check that $\eqref{eq:M=c-matrix}=\eqref{eq:M=c-factorization}\geq
0$. We also check that its square is $K$. By the uniqueness of the square root
it is $M$. ∎

For further reference let us state the formula for the inverse of $M^{-1}$:

$\displaystyle M^{-1}$ $\displaystyle=\begin{bmatrix}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-\frac{1}{2}}&c({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\\\ c^{*}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{-\frac{1}{2}}&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\end{bmatrix}$ (4.14)
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&c\\\
0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-cc^{*})^{\frac{1}{2}}&0\\\ 0&({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c^{*}c)^{-\frac{1}{2}}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ c^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(4.15)

In the real case the operators $c$, $K$ have additional properties:

######  Proposition 4.20.

Suppose that $({\mathcal{W}},Q)$ is a Krein space with conjugation. Let
$S_{1},S_{2}$ be two admissible anti-real involutions on ${\mathcal{W}}$. Let
$K$ and $c$ be defined as in (4.4). Then $K$ is real, $c^{\\#}=c$ and we can
write

$c\coloneqq\Pi_{1}^{(+)}\frac{{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-K}{{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K}\overline{\Pi_{1}^{(+)}}.$ (4.16)

###  4.5 Pseudo-unitary transformations as $\boldsymbol{2\times 2}$ matrices

As in the previous subsection, $S_{1}$ and $S_{2}$ are two admissible
involutions on a Krein space $({\mathcal{W}},Q)$. As explained above,
${\mathcal{W}}_{1}$ and ${\mathcal{W}}_{2}$ denote the space ${\mathcal{W}}$
with the Hilbert structure given by $S_{1}$ resp. $S_{2}$.

######  Proposition 4.21.

Let $R$ be a bounded operator $R$ on ${\mathcal{W}}$. The following are
equivalent:

  1. (1)

$R$ is pseudo-unitary.

  2. (2)

$R$ is invertible and

$\displaystyle R^{*}S_{2}R$ $\displaystyle=S_{1}.$ (4.17)

  3. (3)

$R$ satisfies

$\displaystyle R^{*}S_{2}R$ $\displaystyle=S_{1},$ (4.18a) $\displaystyle
RS_{1}R^{*}$ $\displaystyle=S_{2}.$ (4.18b)

Above, the Hermitian adjoint is understood in the sense of
$R:{\mathcal{W}}_{1}\to{\mathcal{W}}_{2}$.

###### Proof.

(1)$\Rightarrow$(2). Suppose that $R$ is pseudo-unitary.

${(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}R^{*}S_{2}Rw)}_{1}={(Rv\mkern
2.0mu|\mkern 2.0mu\mathopen{}S_{2}Rw)}_{2}=(Rv\mkern 2.0mu|\mkern
2.0mu\mathopen{}QRw)=(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)={(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}S_{1}w)}_{1},$ (4.19)

proves (4.17).

(2)$\Rightarrow$(3). (4.17) and the invertibility of $R$ yields

$\displaystyle R^{-1}=S_{1}R^{*}S_{2},\quad R^{*-1}=S_{2}RS_{1},$ (4.20)
$\displaystyle S_{2}=R^{*-1}S_{1}R^{-1}.$ (4.21)

Inserting (4.20) into (4.21) yields (4.18b).

(3)$\Rightarrow$(1). By (4.18), $S_{1}R^{*}S_{2}$ is the inverse of $R$. Now
we rewrite (4.19) in a different order:

$(Rv\mkern 2.0mu|\mkern 2.0mu\mathopen{}QRw)={(Rv\mkern 2.0mu|\mkern
2.0mu\mathopen{}S_{2}Rw)}_{2}={(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}R^{*}S_{2}Rw)}_{1}={(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}S_{1}w)}_{1}=(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw).$
(4.22)

∎

Every bounded operator $R$ on ${\mathcal{W}}$ can be written as

$R=\begin{bmatrix}R_{++}&R_{+-}\\\ R_{-+}&R_{--}\end{bmatrix},$ (4.23)

with the matrix in the sense of
$R:{\mathcal{Z}}_{1}^{(+)}\oplus{\mathcal{Z}}_{1}^{(-)}\to{\mathcal{Z}}_{2}^{(+)}\oplus{\mathcal{Z}}_{2}^{(-)}$.
If $R$ is pseudo-unitary, its components satisfy various relations:

######  Proposition 4.22.

An operator $R$ on ${\mathcal{W}}$ is pseudo-unitary if and only if

$\displaystyle\begin{aligned}
R^{\mathrlap{*}}{}_{++}R_{++}-R^{\mathrlap{*}}{}_{-+}R_{-+}&={{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}},\quad&R^{\mathrlap{*}}{}_{++}R_{+-}-R^{\mathrlap{*}}{}_{-+}R_{--}&=0,\\\
R^{\mathrlap{*}}{}_{+-}R_{++}-R^{\mathrlap{*}}{}_{--}R_{-+}&=0,\quad&R^{\mathrlap{*}}{}_{+-}R_{+-}-R^{\mathrlap{*}}{}_{--}R_{--}&=-{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}},\end{aligned}$ (4.24) $\displaystyle\begin{aligned}
R_{++}R^{\mathrlap{*}}{}_{++}-R_{+-}R^{\mathrlap{*}}{}_{+-}&={{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}},\quad&R_{++}R^{\mathrlap{*}}{}_{-+}-R_{+-}R^{\mathrlap{*}}{}_{--}&=0,\\\
R_{-+}R^{\mathrlap{*}}{}_{++}-R_{--}R^{\mathrlap{*}}{}_{+-}&=0,\quad&R_{-+}R^{\mathrlap{*}}{}_{-+}-R_{--}R^{\mathrlap{*}}{}_{--}&=-{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}.\end{aligned}$ (4.25)

###### Proof.

Using $S_{1}=\bigl{(}\begin{smallmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
0&-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{smallmatrix}\bigr{)}$ and
$S_{2}=\bigl{(}\begin{smallmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
0&-{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{smallmatrix}\bigr{)}$ with respect
to the decompositions ${\mathcal{Z}}_{1}^{(+)}\oplus{\mathcal{Z}}_{1}^{(-)}$
and ${\mathcal{Z}}_{2}^{(+)}\oplus{\mathcal{Z}}_{2}^{(-)}$, respectively, one
easily sees that (4.18a) is equivalent to (4.24), and (4.18b) is equivalent to
(4.25). ∎

######  Corollary 4.23.

If $R$ is a pseudo-unitary operator on ${\mathcal{W}}$, then

$\displaystyle R_{++}^{*}R_{++}\geq{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}},\quad
R_{++}R_{++}^{*}\geq{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}},$ $\displaystyle
R_{--}^{*}R_{--}\geq{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}},\quad
R_{--}R_{--}^{*}\geq{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}},$

Hence $R_{++}^{-1}$ and $R_{--}^{-1}$ are well-defined.

######  Proposition 4.24.

Suppose that $R$ is a pseudo-unitary operator on ${\mathcal{W}}$. If we set

$\displaystyle c(R)$ $\displaystyle\coloneqq
R_{-+}^{*}R_{--}^{*-1}=R_{++}^{-1}R_{+-},$ (4.26a) $\displaystyle d(R)$
$\displaystyle\coloneqq R_{+-}R_{--}^{-1}=R_{++}^{*-1}R_{-+}^{*},$ (4.26b)

one has the factorizations:

$\displaystyle R$ $\displaystyle=\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&d(R)\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}R_{++}^{*-1}&0\\\
0&R_{--}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
c(R)^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}$ (4.27)
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&0\\\
d(R)^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}R_{++}&0\\\
0&R_{--}^{*-1}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&c(R)\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(4.28)

Moreover, we have the identities

$\displaystyle R_{++}^{*}R_{++}$ $\displaystyle=\big{(}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c(R)c(R)^{*}\big{)}^{-1},$ $\displaystyle
R_{--}^{*}R_{--}=\big{(}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-c(R)^{*}c(R)\big{)}^{-1},$ (4.29) $\displaystyle
R_{++}R_{++}^{*}$ $\displaystyle=\big{(}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-d(R)d(R)^{*}\big{)}^{-1},$ $\displaystyle
R_{--}R_{--}^{*}=\big{(}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-d(R)^{*}d(R)\big{)}^{-1}.$ (4.30)

###### Proof.

The equality of the two formulas for $c(R)$ and $d(R)$ follows from the off-
diagonal relations in (4.24)–(4.25). The decomposition (4.27) can be seen by
multiplying the operator matrices on the right-hand side and using the first
equation of (4.24). ∎

The identity transformation is obviously pseudo-unitary and has the
factorization (4.27), which is described in the following proposition:

######  Proposition 4.25.

The factorization of the identity can be written as

$\displaystyle{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$
$\displaystyle=\begin{bmatrix}\Pi_{2}^{(+)}\Pi_{1}^{(+)}&\Pi_{2}^{(+)}\Pi_{1}^{(-)}\\\
\Pi_{2}^{(-)}\Pi_{1}^{(+)}&\Pi_{2}^{(-)}\Pi_{1}^{(-)}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&d({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}})\\\ 0&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}(\Pi_{1}^{(+)}\Pi_{2}^{(+)})^{-1}&0\\\
0&\Pi_{2}^{(-)}\Pi_{1}^{(-)}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}})^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(4.31)

The operator $c(2,1):{\mathcal{Z}}_{1}^{(-)}\to{\mathcal{Z}}_{1}^{(+)}$
defined in (4.4) coincides with the operator $c({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}})$ from (4.31).

###### Proof.

To see that (4.31) follows from (4.27) it is enough to note that
$(\Pi_{2}^{(+)}\Pi_{1}^{(+)})^{*}=\Pi_{1}^{(+)}\Pi_{2}^{(+)}$. Indeed,

$\displaystyle{(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}\Pi_{2}^{(+)}\Pi_{1}^{(+)}w)}_{2}={(\Pi_{2}^{(+)}v\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Pi_{1}^{(+)}w)}_{2}$
$\displaystyle=(\Pi_{2}^{(+)}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}Q\Pi_{1}^{(+)}w)$ $\displaystyle={(\Pi_{2}^{(+)}v\mkern
2.0mu|\mkern 2.0mu\mathopen{}\Pi_{1}^{(+)}w)}_{1}$
$\displaystyle={(\Pi_{1}^{(+)}\Pi_{2}^{(+)}v\mkern 2.0mu|\mkern
2.0mu\mathopen{}w)}_{1}.$

Now

$\displaystyle\Pi_{2}^{(+)}\Pi_{1}^{(-)}$
$\displaystyle=(\Pi_{2}^{(+)}\Pi_{1}^{(-)}+\Pi_{2}^{(-)}\Pi_{1}^{(+)})\Pi_{1}^{(-)}=\frac{({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-K)}{2}\Pi_{1}^{(-)},$ (4.32)
$\displaystyle\Pi_{2}^{(+)}\Pi_{1}^{(+)}$
$\displaystyle=(\Pi_{2}^{(+)}\Pi_{1}^{(+)}+\Pi_{2}^{(-)}\Pi_{1}^{(-)})\Pi_{1}^{(+)}=\frac{({{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K)}{2}\Pi_{1}^{(+)}.$ (4.33)

Hence, starting from the definition (4.26a), we obtain

$\displaystyle c({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}})$
$\displaystyle=(\Pi_{2}^{(+)}\Pi_{1}^{(+)})^{-1}\Pi_{2}^{(+)}\Pi_{1}^{(-)}$
$\displaystyle=\Pi_{1}^{(+)}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}+K)^{-1}({{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}-K)\Pi_{1}^{(-)}=c(2,1).\qed$

Let us specialize some of the above results to Krein spaces with conjugation.
Suppose that $S_{1}$ and $S_{2}$ are admissible anti-real involutions. Then

$\displaystyle\overline{R_{--}}=R_{++}$ $\displaystyle\eqqcolon p,$
$\displaystyle\overline{R_{-+}}=R_{+-}$ $\displaystyle\eqqcolon q$

and thus every real operator $R$ can be written as

$R=\begin{bmatrix}p&q\\\ \overline{q}&\overline{p}\end{bmatrix},$ (4.34)

in the sense of
$R:{\mathcal{Z}}_{1}^{(+)}\oplus\overline{{\mathcal{Z}}_{1}^{(+)}}\to{\mathcal{Z}}_{2}^{(+)}\oplus\overline{{\mathcal{Z}}_{2}^{(+)}}$.
Here is the real version of Prop. 4.24:

######  Proposition 4.26.

Suppose that $R$ is real pseudo-unitary. The definitions (4.26a) and (4.26b)
can be rewritten as

$\displaystyle c(R)$
$\displaystyle\coloneqq\overline{q}^{*}\overline{p}^{*-1}=p^{-1}q=c(R)^{\\#},$
(4.35a) $\displaystyle d(R)$ $\displaystyle\coloneqq
q\overline{p}^{-1}=p^{*-1}\overline{q}^{*}=d(R)^{\\#},$ (4.35b)

and one has the factorization:

$R=\begin{bmatrix}{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}&d(R)\\\ 0&{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}\end{bmatrix}\begin{bmatrix}p^{*-1}&0\\\
0&\overline{p}\end{bmatrix}\begin{bmatrix}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}&0\\\ c(R)^{*}&{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\end{bmatrix}.$
(4.36)

###  4.6 Pseudo-unitary generators

Let $({\mathcal{W}},Q)$ be a pre-pseudo-unitary space.

######  Definition 4.27.

We say that a densely defined operator $B$ on ${\mathcal{W}}$ _infinitesimally
preserves $Q$_ if $B$ is a generator of a one-parameter group
$\mathrm{e}^{-\mathrm{i}tB}$ on ${\mathcal{W}}$ and

$(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}QBw)=(Bv\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qw),\quad v,w\in\operatorname{{\mathcal{D}}}(B).$ (4.37)

If in addition $Q$ is non-degenerate, then we will say that $B$ is a _pseudo-
unitary generator_. The quadratic form defined by (4.37) will be called the
_energy_ or _Hamiltonian quadratic form of $B$ on
$\operatorname{{\mathcal{D}}}(B)$_.

######  Proposition 4.28.

Let $B$ be a generator of a one-parameter group on ${\mathcal{W}}$. Then
$\mathrm{e}^{-\mathrm{i}tB}$, $t\in{{\mathbb{R}}}$, preserves $Q$ if and only
if $B$ infinitesimally preserves $Q$.

###### Proof.

Let us show $\Leftarrow$. Assume first that
$v,w\in\operatorname{{\mathcal{D}}}(B)$. Then

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}(\mathrm{e}^{-\mathrm{i}tB}v|Q\mathrm{e}^{-\mathrm{i}tB}w)$
$\displaystyle=$
$\displaystyle\mathrm{i}(B\mathrm{e}^{-\mathrm{i}tB}v|Q\mathrm{e}^{-\mathrm{i}tB}w)-\mathrm{i}(\mathrm{e}^{-\mathrm{i}tB}v|QB\mathrm{e}^{-\mathrm{i}tB}w)\,=\,0.$
(4.38)

Hence

$(\mathrm{e}^{-\mathrm{i}tB}v|Q\mathrm{e}^{-\mathrm{i}tB}w)=(v|Qw).$ (4.39)

By the density of $\operatorname{{\mathcal{D}}}(B)$ and the boundedness of $Q$
and $\mathrm{e}^{-\mathrm{i}tB}$, (4.39) extends to the whole ${\mathcal{W}}$.

In the proof of the $\Rightarrow$ we use the above arguments in the reverse
order (with the exception of the density argument, which is not needed). ∎

The following proposition describes a large class of pseudo-unitary
transformations and pseudo-unitary generators on Krein spaces.

######  Proposition 4.29.

Suppose that $({\mathcal{W}},Q)$ is a Krein space and $S_{\bullet}$ is an
admissible involution with the corresponding scalar product $(\cdot\mkern
2.0mu|\mkern 2.0mu\mathopen{}\cdot)_{\bullet}$. If $B$ is a densely defined
operator on ${\mathcal{W}}$, self-adjoint in the sense of
${\mathcal{W}}_{\bullet}$ and commuting with $S_{\bullet}$, then $B$ is a
pseudo-unitary generator on $({\mathcal{W}},Q)$ in the sense of Def. 4.27.

###### Proof.

Clearly, $\mathrm{e}^{-\mathrm{i}tB}$ is a unitary operator on
${\mathcal{W}}_{\bullet}$ commuting with $S_{\bullet}$. Therefore, it is a
pseudo-unitary transformation. ∎

######  Definition 4.30.

A densely defined operator $B$ on a Krein space $({\mathcal{W}},Q)$ is called
a _stable pseudo-unitary generator_ if it is similar to self-adjoint,
$\operatorname{{\mathcal{N}}}(B)=\\{0\\}$, and $\operatorname{sgn}(B)$ is an
admissible involution. $B$ is called a _strongly stable pseudo-unitary
generator_ if in addition it is invertible.

In other words, a stable pseudo-unitary generator has a positive Hamiltonian
and a strongly stable generator has a positive Hamiltonian bounded away from
zero.

###  4.7 Evolutions preserving a Hermitian form

Suppose that $({\mathcal{W}},Q)$ is a pre-pseudo-unitary space.

######  Theorem 4.31.

Suppose that all assumptions of Thm. 3.12 are satisfied. In addition, assume
that for all $t\in I$ the operators $B(t)$ infinitesimally preserve $Q$. Then
the evolution $R(t,s)$ on ${\mathcal{W}}$ preserves $Q$.

###### Proof.

Recall that among the assumptions of Thm. 3.12 there is the existence of a
Hilbertizable space ${\mathcal{V}}$ densely and continuously embedded in
${\mathcal{W}}$ such that
${\mathcal{V}}\subset\operatorname{{\mathcal{D}}}(B(t))$. Besides, for any
$w\in{\mathcal{V}}$ this theorem implies that

$\displaystyle\mathrm{i}\partial_{t}R(t,s)w$ $\displaystyle=B(t)R(t,s)w.$
(4.40)

Hence the above theorem follows by repeating the proof of Prop. 4.28, where we
use ${\mathcal{V}}$ instead of $\operatorname{{\mathcal{D}}}(B)$. ∎

###  4.8 Bisolutions and inverses

Let $({\mathcal{W}},Q)$ be a Krein space. Then naturally
$L^{2}(I,{\mathcal{W}})$ is also a Krein space with the Hermitian form

$(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)\coloneqq\int_{I}\big{(}v(t)\mkern
2.0mu\big{|}\mkern 2.0mu\mathopen{}Qw(t)\big{)}\mathop{}\\!\mathrm{d}t,$
(4.41)

and compatible scalar products

$(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}w)_{\bullet}\coloneqq\int_{I}\big{(}v(t)\mkern
2.0mu\big{|}\mkern
2.0mu\mathopen{}w(t)\big{)}_{\bullet}\mathop{}\\!\mathrm{d}t.$ (4.42)

Let $\\{R(t,s)\\}_{t,s\in I}$ be a strongly continuous pseudo-unitary
evolution , on $({\mathcal{W}},Q)$. We denote by $B(t)$ the (heuristic)
generator of $R(t,s)$. Recall that in Sect. 3.3 we considered the (heuristic)
evolution equation operator $M=\partial_{t}+\mathrm{i}B(t)$. Note that
$M=\partial_{t}+\mathrm{i}B(t)$ is (heuristically) anti-$Q$-Hermitian on
$L^{2}(I,W)$. We will give a rigorous version of this statement a little
later, in (4.50).

In Sect. 3.3 we introduced various inverses and bisolutions of $M$. In this
subsection we add the Krein structure to the picture.

First, as in Subsect. 3.3, we define the _Pauli–Jordan bisolution_
$E^{\mathrm{PJ}}$, and the _forward and backward inverses_ $E^{\vee}$, resp.
$E^{\wedge}$.

######  Proposition 4.32.

$E^{\mathrm{PJ}}$ is $Q$-Hermitian and the $Q$-adjoint of $E^{\vee}$ is
contained in $-E^{\wedge}$. More precisely, for $v,w\in
C_{\mathrm{c}}\bigl{(}I,{\mathcal{W}}\bigr{)}$ we have

$\displaystyle(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}QE^{\mathrm{PJ}}w)$
$\displaystyle=(E^{\mathrm{PJ}}v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw),$
$\displaystyle(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}QE^{\wedge}w)$
$\displaystyle=-(E^{\vee}v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw).$

Consider now non-classical bisolutions and inverses. As in Subsect. 3.4, for
finite $t_{-},t_{+}$ we consider
$({\mathcal{Z}}_{+}^{(+)},{\mathcal{Z}}_{-}^{(-)})$, a pair of closed
subspaces of ${\mathcal{W}}$. We set

${\mathcal{Z}}_{-}^{(+)}:={\mathcal{Z}}_{-}^{(-)\perp
Q},\quad{\mathcal{Z}}_{+}^{(-)}:={\mathcal{Z}}_{+}^{(+)\perp Q}.$ (4.43)

As in (3.17) we define
${\mathcal{Z}}_{\mp}^{(-)}(t),{\mathcal{Z}}_{\pm}^{(+)}(t)$, so that they
satisfy (3.18).

If $t_{\pm}=\pm\infty$, we suppose that
$\big{(}{\mathcal{Z}}_{\pm}^{(+)}(t),{\mathcal{Z}}_{\mp}^{(-)}(t)\big{)}$ are
two families of subspaces satisfying (3.18), and in addition

${\mathcal{Z}}_{-}^{(+)}(t)={\mathcal{Z}}_{-}^{(-)}(t)^{\perp
Q},\quad{\mathcal{Z}}_{+}^{(-)}(t)={\mathcal{Z}}_{+}^{(+)}(t)^{\perp Q}.$
(4.44)

Clearly, if (4.44) holds for one $t\in I^{\mathrm{cl}}$, it holds for all
$t\in I^{\mathrm{cl}}$.

By Prop. 4.11 (2), the pair
$\big{(}{\mathcal{Z}}_{+}^{(+)}(t),{\mathcal{Z}}_{-}^{(-)}(t)\big{)}$
satisfies the asymptotic complementarity condition iff the pair
$\big{(}{\mathcal{Z}}_{+}^{(-)}(t),{\mathcal{Z}}_{-}^{(+)}(t)\big{)}$ does.
Assume that this is the case. We define the corresponding inverses $E^{(+-)}$
and $E^{(-+)}$ considered as operators $C_{\mathrm{c}}(I,{\mathcal{W}})\to
C(I,{\mathcal{W}})$, as in Def. 3.5.

######  Proposition 4.33.

The $Q$-adjoint of $E^{(+-)}$ is contained in $-E^{(-+)}$. More precisely, for
$v,w\in C_{\mathrm{c}}\bigl{(}I,{\mathcal{W}}\bigr{)}$ we have

$(E^{(+-)}w\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qv)=-(w\mkern 2.0mu|\mkern
2.0mu\mathopen{}QE^{(-+)}v).$ (4.45)

###### Proof.

Note that

$R(s,t)^{*Q}=R(t,s),\quad\Lambda^{(\boxplus-)}(t)^{*Q}=\Lambda^{(-\boxplus)}(t),\quad\Lambda^{(+\boxminus)}(t)^{*Q}=\Lambda^{(\boxminus+)}(t).$

Now,

$\displaystyle(E^{(+-)}w\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qv)=$
$\displaystyle\int_{I}\big{(}w(t)|QE^{(+-)}(s,t)^{*Q}v(s)\big{)}\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}s,$
$\displaystyle E^{(+-)}(s,t)^{*Q}$ $\displaystyle=$
$\displaystyle\theta(s-t)R(s,t)^{*Q}\Lambda^{(\boxplus-)}(t)^{*Q}-\theta(t-s)R(s,t)^{*Q}\Lambda^{(+\boxminus)}(t)^{*Q}$
$\displaystyle=$
$\displaystyle\theta(s-t)R(t,s)\Lambda^{(-\boxplus)}(t)-\theta(t-s)R(t,s)\Lambda^{(\boxminus+)}(t)\,=\,-E^{(-+)}(t,s).\qed$

Suppose now that the subspaces ${\mathcal{Z}}_{+}^{(+)}(t)$,
${\mathcal{Z}}_{-}^{(+)}(t)$ are m-positive. (Clearly, if this holds for one
$t\in I^{\mathrm{cl}}$, then it is true for all $t\in I^{\mathrm{cl}}$).
Equivalently, we can assume that ${\mathcal{Z}}_{+}^{(-)}$,
${\mathcal{Z}}_{-}^{(-)}$ are m-negative. In other words, we suppose that we
have chosen two admissible involutions $S_{+}$ and $S_{-}$, so that
${\mathcal{Z}}_{\pm}^{(+)}$ are their positive subspaces and
${\mathcal{Z}}_{\pm}^{(-)}$ their negative subspaces. We define the
bisolutions $E_{\pm}^{(+)}$ and $E_{\pm}^{(-)}$ as in Def. 3.7 as operators
$C_{\mathrm{c}}(I,{\mathcal{W}})\to C(I,{\mathcal{W}})$.

######  Proposition 4.34.

$E_{\pm}^{(+)}$ and $E_{\pm}^{(-)}$ are $Q$-Hermitian and $Q$-positive. More
precisely, for $v,w\in C_{\mathrm{c}}(I,{\mathcal{W}})$ we have

$\displaystyle(E_{\pm}^{(+)}w\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qv)$
$\displaystyle=(w\mkern 2.0mu|\mkern 2.0mu\mathopen{}QE_{\pm}^{(+)}v),$
$\displaystyle\quad(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}QE_{\pm}^{(+)}v)$
$\displaystyle\geq 0,$ $\displaystyle(E_{\pm}^{(-)}w\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qv)$ $\displaystyle=(w\mkern 2.0mu|\mkern
2.0mu\mathopen{}QE_{\pm}^{(-)}v),$ $\displaystyle\quad(v\mkern 2.0mu|\mkern
2.0mu\mathopen{}QE_{\pm}^{(-)}v)$ $\displaystyle\geq 0.$

What is more remarkable, under the present assumptions asymptotic
complementarity holds automatically for both
$\big{(}{\mathcal{Z}}_{-}^{(-)}(t),{\mathcal{Z}}_{+}^{(+)}(t)\big{)}$ and
$\big{(}{\mathcal{Z}}_{-}^{(+)}(t),{\mathcal{Z}}_{+}^{(-)}(t)\big{)}$:

######  Theorem 4.35.

For any $t\in I^{\mathrm{cl}}$, the two pairs of subspaces of ${\mathcal{W}}$

$\bigl{(}{\mathcal{Z}}_{-}^{(+)}(t),{\mathcal{Z}}_{+}^{(-)}(t)\bigr{)}\quad\text{and}\quad\bigl{(}{\mathcal{Z}}_{+}^{(+)}(t),{\mathcal{Z}}_{-}^{(-)}(t)\bigr{)},$
(4.46)

are complementary. Thus we can define the operators $E^{(+-)}$, $E^{(-+)}$.

###### Proof.

A pair consisting of an m-positive and m-negative subspace is always
complementary by Prop. 4.17. ∎

In QFT the above construction is used in the following situation. Suppose that
$I=\mathbb{R}$,

$B(\pm\infty):=\lim_{t\to\pm\infty}B(t)$ (4.47)

exist and are stable pseudo-unitary generators (see Def. 4.30). Then we assume
that

$\displaystyle\lim_{t\pm\infty}{\mathcal{Z}}_{\pm}^{(+)}(t)$
$\displaystyle=\operatorname{{\mathcal{R}}}\big{(}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}_{[0,\infty[}\big{(}B(\pm\infty)\big{)}\big{)},$
$\displaystyle\lim_{t\pm\infty}{\mathcal{Z}}_{\pm}^{(-)}(t)$
$\displaystyle=\operatorname{{\mathcal{R}}}\big{(}{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}}_{]-\infty,0]}\big{(}B(\pm\infty)\big{)}\big{)}.$ (4.48)

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

######  Definition 4.36.

With the choice (4.48), the bisolutions $E_{\pm}^{(+)}$ are called the _in/out
positive frequency bisolutions_ , the bisolutions $E_{\pm}^{(-)}$ are called
the _in/out negative frequency bisolutions_ , and the inverses $E^{(+-)}$,
resp. $E^{(-+)}$ are called the _in-out Feynman_ , resp. the _in-out anti-
Feynman inverse_. They are often denoted $E^{\mathrm{F}}$, resp.
$E^{\overline{\mathrm{F}}}$.

###  4.9 Well-posed realizations of the evolution equation operator

Suppose that $({\mathcal{W}},Q)$ is a Krein space and $I\ni t\mapsto B(t)$ is
a family of pseudo-unitary generators satisfying the assumptions of Thm. 3.12.
By Thm 4.31 we obtain a pseudo-unitary evolution $R(t,s)$ on ${\mathcal{W}}$
satisfying

$\displaystyle\mathrm{i}\partial_{t}R(t,s)w$ $\displaystyle=B(t)R(t,s)w,\quad
w\in{\mathcal{V}},$ (4.49)

for a certain Hilbertizable space ${\mathcal{V}}$ densely and continuously
embedded in ${\mathcal{W}}$. We can now propose a rigorous meaning for the
anti-$Q$-Hermiticity of the operator $M=\partial_{t}+\mathrm{i}B(t)$ on the
Krein space $L^{2}(I,{\mathcal{W}})$: for $v,w\in
C_{\mathrm{c}}(I,{\mathcal{V}})\cap C_{\mathrm{c}}^{1}(I,{\mathcal{W}})$,

$\Big{(}w\mkern 2.0mu\Big{|}\mkern
2.0mu\mathopen{}Q\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}v\Big{)}=-\Big{(}\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}w\mkern
2.0mu\Big{|}\mkern 2.0mu\mathopen{}Qv\Big{)}.$ (4.50)

For the remaining part of this subsection we assume that $I$ is finite.

######  Proposition 4.37.

Suppose that the pair $({\mathcal{Z}}_{-}^{(-)},{\mathcal{Z}}_{+}^{(+)})$
satisfies the asymptotic complementarity, so that we can define $E^{(+-)}$ and
$E^{(-+)}$. Then they extend to bounded operators on $L^{2}(I,{\mathcal{W}})$,
their ranges are dense and their null-spaces are $\\{0\\}$.

###### Proof.

The boundedness is obvious. By Proposition 3.14 (1) for any $v\in
C_{\mathrm{c}}(I,{\mathcal{V}})\cap C_{\mathrm{c}}^{1}({\mathcal{W}})$ we have

$E^{(+-)}\bigl{(}\partial_{t}+\mathrm{i}B(t)\bigr{)}v=v.$ (4.51)

Hence ${\mathcal{R}}(E^{(+-)})$ contains $C_{\mathrm{c}}(I,{\mathcal{V}})\cap
C_{\mathrm{c}}^{1}(I,{\mathcal{W}})$, which is dense in
$L^{2}(I,{\mathcal{W}})$. The same argument shows that
${\mathcal{R}}(E^{(-+)})$ is dense in $L^{2}(I,{\mathcal{W}})$. Now

${\mathcal{N}}(E^{(+-)})={\mathcal{R}}(E^{{(+-)}*Q})^{\perp
Q}={\mathcal{R}}(E^{{(-+)}})^{\perp Q}=\\{0\\}.$

∎

Thus by Prop. 4.37, for finite $I$ and under the asymptotic complementarity
assumption we can define operators with dense domains

$M^{(+-)}:=(E^{(+-)})^{-1},\quad M^{(-+)}:=(E^{(-+)})^{-1}.$

They satisfy

$(M^{(+-)})^{*Q}=M^{(-+)}$

and $0$ belongs to their resolvent set.

###  4.10 Nested pre-pseudo-unitary pairs

The construction of the Krein space that we will use to study the Klein-Gordon
equation involves an interpolation from a nested pair of Hilbertizable spaces
equipped with a Hermitian pairing. Let us describe this simple construction in
detail. In the next subsection we will describe evolutions on such nested
pairs.

######  Definition 4.38.

Let $\beta>0$. A _nested pre-pseudo-unitary pair_
$({\mathcal{W}}_{-\beta},{\mathcal{W}}_{\beta},Q)$ consists of a pair of
Hilbertizable spaces ${\mathcal{W}}_{-\beta}$, ${\mathcal{W}}_{\beta}$, where
${\mathcal{W}}_{\beta}$ is densely and continuously embedded in
${\mathcal{W}}_{-\beta}$ and a Hermitian pairing, that is a sesquilinear form

${\mathcal{W}}_{\beta}\times{\mathcal{W}}_{-\beta}\ni(v,w)\mapsto(v\mkern
2.0mu|\mkern 2.0mu\mathopen{}Qw)\in{{\mathbb{C}}},$ (4.52)

which is Hermitian on ${\mathcal{W}}_{\beta}$, i.e.,

$(v\mkern 2.0mu|\mkern 2.0mu\mathopen{}Qw)=\overline{(w\mkern 2.0mu|\mkern
2.0mu\mathopen{}Qv)},\quad v,w\in{\mathcal{W}}_{\beta},$ (4.53)

and bounded, i.e., for some (hence all) compatible norms