# On generally covariant mathematical formulation of Feynman integral in
Lorentz signature

András László  Wigner Research Centre for Physics, Budapest
<EMAIL_ADDRESS>
###### Abstract

It is widely accepted that the Feynman integral is one of the most promising
methodologies for defining a generally covariant formulation of
nonperturbative interacting quantum field theories (QFTs) without a fixed
prearranged causal background. Recent literature suggests that if the
spacetime metric is not fixed, e.g. because it is to be quantized along with
the other fields, one may not be able to avoid considering the Feynman
integral in the original Lorentz signature, without Wick rotation. Several
mathematical phenomena are known, however, which are at some point
showstoppers to a mathematically sound definition of Feynman integral in
Lorentz signature. The Feynman integral formulation, however, is known to have
a differential reformulation, called to be the master Dyson–Schwinger (MDS)
equation for the field correlators. In this paper it is shown that a
particular presentation of the MDS equation can be cast into a mathematically
rigorously defined form: the involved function spaces and operators can be
strictly defined and their properties can be established. Therefore, MDS
equation can serve as a substitute for the Feynman integral, in a
mathematically sound formulation of constructive QFT, in arbitrary signature,
without a fixed background causal structure. It is also shown that even in
such a generally covariant setting, there is a canonical way to define the
Wilsonian regularization of the MDS equation. The main result of the paper is
a necessary and sufficient condition for the regularized MDS solution space to
be nonempty, for conformally invariant Lagrangians. This theorem also provides
an iterative approximation algorithm for obtaining regularized MDS solutions,
and is guaranteed to be convergent whenever the solution space is nonempty.
The algorithm could eventually serve as a method for putting Lorentz signature
QFTs onto lattice, in the original metric signature.

Keywords: Feynman integral, master Dyson–Schwinger equation, generally
covariant, constructive field theory

##  1 Introduction

By now, a lot is known about the mathematically sound formulation of
interacting quantum field theory (QFT), using perturbation theory [1].
However, still until now, there is no widely accepted concise mathematical
formulation known for nonperturbative interacting QFT. Strictly speaking, as
of now, it is only conjectured that eventually one could well-define an
interacting QFT model in a nonperturbative manner, in a constructive way, e.g.
as specified by a Lagrangian. A well known promising attempt for the
nonperturbative approach is the algebraic quantum field theory (AQFT) [2, 3].
AQFT is known to capture several important qualitative aspects of the QFT
formalism in physics, such as the spin-statistics theorem, but there are no
known concrete AQFT constructions in the compexity of e.g. a 3+1 dimensional
full quantum electrodynamics. Concrete AQFT models, as of now, are only known
for free particles in arbitrary dimensions, or for simple systems, such as
discrete Ising models in 1+1 dimensional and discrete spacetimes, or for
particular simple systems in spacetime dimensions typically lower than 3+1.
There are also recent advances of perturbative AQFT on causal sets, in which
framework concrete interacting models are constructed by now [4], assuming a
finite system of causal sets. Due to the difficulties of nonperturbative
formulation, the perturbative rigorous formulation of constructive QFT (pQFT)
was seriously considered by a number of authors [5, 6, 7, 8, 9]. In
particular, [10] proves the perturbative renormalizability of Yang–Mills
interactions over globally hyperbolic spacetimes. Moreover, a generally
covariant framework was already developed [11]. However, it is generally
thought that the only promising framework, which could be capable of
formalizing _nonperturbative_ interacting generally covariant QFT models in
the continuum limit, is likely to be the Feynman integral formulation [12].

A lot is known about Feynman intergrals [13, 14], but in Lorentz signature,
without taking a Wick rotation, it is seems to be still not a completely
understood mathematical construction, although the modern literature seems to
tighten the noose on the measure theoretically well defined Feynman integral
[15, 16]. Other authors [17] argue, that Feynman integral should not be,
strictly speaking, understood in the measure theoretical sense, i.e. in the
sense of infinitesimal summation, but in a more generalized sense. That kind
of picture is indeed supported by the fact that e.g. for a fermionic system,
the Feynman integral is defined as a Berezin integral, which indeed has little
link with integration in terms of infinitesimal summation. To complicate the
picture, recent literature suggests [10, 18, 19, 20] that in order to apply
Feynman integral formalism to a generally covariant setting, in which case the
a background spacetime metric is not fixed, the applicability of the usual
Wick rotation from Lorentzian to Euclidean signature can be problematic.

The above issues with the Lorentz signature Feynman integral formulation can
be circuimvented using the well known differential reformulation of Feynman
integral formalism, called to be the master Dyson–Schwinger (MDS) equations
for the field correlators (see e.g. [21] for a didactic review). From the
usually presented form of the MDS equation in the QFT literature, it is not
immediately evident that the function spaces and operators involved in the MDS
equation are well defined, and are not merely symbolical summaries of
heuristic QFT protocols. In this paper, however, it is argued that with the
right choice of variables, these objects can be made mathematically well
defined, and as such, the MDS equation can be used to substitute the Feynman
integral for a mathematically sound definition of constructive nonperturbative
generally covariant QFT. It will be also shown, that in these variables the
Wilsonian regularized version of the MDS equation can also be canonically
defined in a generally covariant setting, which is not yet described in the
literature. The main result of the paper is a theorem about a necessary and
sufficient condition for the regularized MDS equation to have nonempty
solution space, for theories with classically conformally invariant
Lagrangians. The pertinent theorem is constructive in the sense that it
provides a (probably slowly converging) iterative algorithm for approximating
MDS solutions, which is guaranteed to be convergent whenever the solution
space is nonempty. This method can eventually be also employed for doing
lattice QFT-like calculations in arbitrary signatures, in particular, in the
original Lorentz signature.

The structure of the paper is as follows. In Section 2 the heuristic form of
the MDS equation is recalled, as derived from the heuristic Feynman integral
formulation in usual QFT. The rest of the paper intends to keep mathematical
rigor. In Section 3 the function spaces and operators needed to define the
(unregularized) MDS operator are presented. In Section 4, these are
generalized in the distributional sense, and the Wilsonian regularized version
of the MDS equation is invoked and justified. Section 5 is dedicated for the
main theorem of the paper about a necesary and sufficient condition for the
existence of solutions of the regularized MDS equation, for conformally
invariant Lagrangians. A was added in order to pin down the precise continuity
properties of a typical Euler–Lagrange functional in a standard classical
field theory, which is key in the construction. B was added for completeness,
in order to define the Wilsonian renormalizability in a generally covariant
way, given the notion of Wilsonian regularization. The paper heavily relies on
the theory of non-normable topological vector spaces (TVS), and therefore a
supplementary material [22] is provided for a recollection of important and
sometimes counterintuitive theorems on these, for readers not specialized in
the theory of TVS.

##  2 Feynman integral and the heuristic form of the MDS equation

We briefly recall the justification of the MDS equation in the Feynman
integral formulation of QFT. Let $F$ denote the space of all (that is, off-
shell) smooth classical field configurations. As expanded in A, in most models
it is safe to assume that $F$ is a topological affine space, such that its
subordinate vector space $\mathbb{F}$, the space of smooth field variations,
carryies a nuclear Fréchet topology. The affineness of $F$ is necessary in
order to naturally accomodate gauge fields. $\mathbb{F}^{*}$ will denote the
topological dual of $\mathbb{F}$, understood with the standard strong dual
topology. In the Feynman integral formulation of QFT, it is postulated that
the evaluation method for Feynman type (i.e., causally ordered) quantum vacuum
expectation value of observables in a (not necessarily unique) vacuum state
$\rho$ is the following. Given a fixed reference field $\psi_{0}\in F$ and
test functionals $J_{1},\dots,J_{n}\in\mathbb{F}^{*}$, the causally ordered
quantum vacuum expectation value of the polynomial observable
$(J_{1}|\cdot{-}\psi_{0})\cdots(J_{n}|\cdot{-}\psi_{0}):\,F\rightarrow\mathbb{R}$
is declared to be

$\displaystyle\int\limits_{\psi\in
F}(J_{1}|\psi{-}\psi_{0})\cdots(J_{n}|\psi{-}\psi_{0})\;\;\rme^{\mathrm{i}\frac{1}{\hbar}S(\psi)}\;\mathrm{d}\rho(\psi)\;\Bigg{/}\;\int\limits_{\psi\in
F}\rme^{\mathrm{i}\frac{1}{\hbar}S(\psi)}\;\mathrm{d}\rho(\psi),$ (1)

where the symbol $\mathrm{d}\rho(\cdot)$ denotes the hypothetical Feynman
measure corresponding to a vacuum state $\rho$, $(\cdot|\cdot)$ denotes the
duality pairing form between $\mathbb{F}^{*}$ and $\mathbb{F}$, whereas
$S:\,F\rightarrow\mathbb{R}$ is the action functional of the underlying
classical field theory. In the heuristic calculations, $\mathrm{d}\rho(\cdot)$
is handled as if it were a Lebesgue measure on $F$, and as if
$\rme^{\mathrm{i}\frac{1}{\hbar}S(\cdot)}\,\mathrm{d}\rho(\cdot)$ were a
finite measure, having finite moments and analytic Fourier transform. A sign
change $\hbar\mapsto{-}\hbar$ would correspond to a reversal in the causal
ordering, if there were any _a priori_ causal structure over the spacetime
manifold (which in fact, is not needed to be assumed at this point). The
hypotethical _partition function_ condenses all these information about the
state $\rho$, and would be a mapping

$\displaystyle
Z_{\hbar,\psi_{0}}:\quad\mathbb{F}^{*}\rightarrow\mathbb{C},\quad J\mapsto
Z_{\hbar,\psi_{0}}(J):=\int\limits_{\psi\in
F}\rme^{\mathrm{i}\,\left(J|\psi{-}\psi_{0}\right)}\;\rme^{\mathrm{i}\frac{1}{\hbar}S(\psi)}\;\mathrm{d}\rho(\psi),$
(2)

i.e. the formal Fourier transform of the hypothetical measure
$\rme^{\mathrm{i}\frac{1}{\hbar}S(\cdot)}\,\mathrm{d}\rho(\cdot)$. The
collection of _$n$ -field correlators_

$\displaystyle G_{\hbar,\psi_{0}}^{(n)}$ $\displaystyle:=$
$\displaystyle\left.\left((-\mathrm{i})^{n}\,\frac{1}{Z_{\hbar,\psi_{0}}(J)}\;D^{(n)}Z_{\hbar,\psi_{0}}(J)\right)\right|_{J=0}$
(3)

is an other means to rephrase these information about the state $\rho$, and
also can be used to evaluate the quantum expectation values Eq.(1) by simple
duality pairing, like
$\big{(}J_{1}{\otimes}{\dots}{\otimes}J_{n}\,\big{|}\,G_{\hbar,\psi_{0}}^{(n)}\big{)}$.
Here $D^{(n)}Z_{\hbar,\psi_{0}}$ is assumed to behave like the $n$-th Fréchet
derivative of the partition function $J\mapsto Z_{\hbar,\psi_{0}}(J)$,
implicitly assuming that $Z_{\hbar,\psi_{0}}$ is $n$-times continuously
Fréchet differentiable (and for fermion fields, this differentiation is
assumed to be a graded differentiation). Since the partition function would be
a map $Z_{\hbar,\psi_{0}}:\mathbb{F}^{*}\rightarrow\mathbb{C}$, the collection
of field correlators
$G_{\hbar,\psi_{0}}:=\big{(}G_{\hbar,\psi_{0}}^{(0)},G_{\hbar,\psi_{0}}^{(1)},{\dots},G_{\hbar,\psi_{0}}^{(n)},{\dots}\big{)}$
would sit in
$\mathcal{T}(\mathbb{F}):=\mathop{\bigoplus}\limits_{n\in\mathbb{N}_{0}}\mathop{\otimes}\limits^{n}\mathbb{F}$,
i.e. in the tensor algebra of $\mathbb{F}$, or more precisely in a graded-
symmetrized subspace of $\mathcal{T}(\mathbb{F})$.

Let $E(\psi):=D_{F}S(\psi)$ denote the Euler–Lagrange functional, i.e. the
derivative of the action functional $S$, evaluated at the classical field
configuration $\psi\in F$. It would be a map
$E:\,F\times\mathbb{F}\rightarrow\mathbb{R},\,(\psi,\delta\\!\psi)\mapsto\left(E(\psi)\big{|}\delta\\!\psi\right):=\left(D_{F}S(\psi)\big{|}\delta\\!\psi\right)$,
being linear in its second variable, since it is a derivative. In the usual
QFT protocol it is assumed that the EL functional $E$ is multipolynomial, and
thus so is the real valued map
$\psi\mapsto\left(E(\psi)\,\big{|}\,\delta\\!\psi\right)$ for any fixed field
variation $\delta\\!\psi\in\mathbb{F}$. Let
$\mathbf{E}((-\mathrm{i})D_{\mathbb{F}^{*}}+\psi_{0})$ be the multipolynomial
differential operator defined by the polynomial coefficients of the Euler-
Lagrange functional $E$. Applying the usual rules of formal Fourier transform,
a function $Z:\mathbb{F}^{*}\rightarrow\mathbb{C}$ is of the form Eq.(2), up
to a complex multiplyer, if and only if it satisfies the master
Dyson–Schwinger (MDS) equation

$\displaystyle\left.\Big{(}\;\mathbf{E}((-\mathrm{i})D_{\mathbb{F}^{*}}+\psi_{0})\;Z\;\Big{)}\right|_{J}$
$\displaystyle=$ $\displaystyle-\hbar\,J\;Z(J)\qquad\qquad\qquad(\forall
J\in\mathbb{F}^{*}),$ (4)

see e.g. [21] for a didactic derivation. The operational meaning of this usual
presentation of the MDS equation might not seem immediately evident. However,
expressing $Z_{\hbar,\psi_{0}}$ via its formal Taylor series, encoded by the
collection of field correlators
$G_{\hbar,\psi_{0}}\in\mathcal{T}(\mathbb{F})$, the MDS equation Eq.(4) is
seen to be equivalent to

$\displaystyle\mathrm{we\;search\;for}\;G\in\mathcal{T}(\mathbb{F})\;\mathrm{such\;that}:$
(5)
$\displaystyle\Big{.}G^{(0)}=1\quad\mathrm{and}\quad\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi)}\,G=\mathrm{i}\,\hbar\,L_{\delta\\!\psi}\,G\qquad(\forall\,\delta\\!\psi\in\mathbb{F}).$
(6)

The symbols of this equation mean the following. $L_{\delta\\!\psi}$ denotes
the left-multiplication operator in the tensor algebra
$\mathcal{T}(\mathbb{F})$ by the one-vector $\delta\\!\psi\in\mathbb{F}$. The
symbol $\text{\Large$\iota$}_{p}$ denotes the left-insertion operator by some
element $p$ from the topological dual space of $\mathcal{T}(\mathbb{F})$. The
map $E_{\psi_{0}}:\,\mathbb{F}\rightarrow\mathbb{F}^{*}$ is defined via
$E_{\psi_{0}}:=E\circ(\mathrm{I}_{\mathbb{F}}+\psi_{0})$ from the original
Euler–Lagrange functional $E:\,F\rightarrow\mathbb{F}^{*}$, i.e. it is the
Euler–Lagrange functional with respect to a fixed reference field $\psi_{0}\in
F$, re-expressed on the space of field variations $\mathbb{F}$. Since it was
assumed to be multipolynomial, it can eventually be regarded as a linear map
$E_{\psi_{0}}:\,\mathcal{T}(\mathbb{F})\rightarrow\mathbb{F}^{*}$. As such, it
may be identified with an element
$\mathbf{E}_{\psi_{0}}\in\big{(}\mathcal{T}(\mathbb{F})\big{)}^{*}\otimes\mathbb{F}^{*}$,
and correspondingly $(E_{\psi_{0}}|\delta\\!\psi)$ with
$(\mathbf{E}_{\psi_{0}}|\delta\\!\psi)\in\big{(}\mathcal{T}(\mathbb{F})\big{)}^{*}\,$
($\forall\,\delta\\!\psi\in\mathbb{F}$), which then has the corresponding
left-insertion operator
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi)}$ acting over
$\mathcal{T}(\mathbb{F})$. The spaces and operators involved in Eq.(6) would
be perfectly meaningful if the space of fields $F$ were finite dimensional,
and could be used as a substitute for Feynman integral formulation Eq.(1),
regardless of e.g. a metric signature or other auxiliary information on the
details of the underlying classical theory described by the Euler–Lagrange
functional $E$. In Section 3 it shall be shown that the pertinent objects can
be made well-defined even when $F$ is indeed the infinite dimensional space of
smooth off-shell field configurations in a realistic field theory. The Eq.(6)
presentation of the MDS equation does not seem to be described in the
literature.

In QFT, it is also necessary to consider the Wilsonian regularized version of
the Feynman integral. Wilsonian regularization means performing the Feynman
integral Eq.(1) on a subspace of off-shell fields with their high frequency
modes suppressed. In a generally covariant setting the meaning of this might
not seem immediately evident, but Wilsonian regularized Feynman type
expectation value of the observable
$(J_{1}|\cdot{-}\psi_{0})\cdots(J_{n}|\cdot{-}\psi_{0}):\,F\rightarrow\mathbb{R}$
can be postulated as

$\displaystyle\int\limits_{\delta\\!\psi\in\mathcal{R}[\mathbb{F}]}(J_{1}|\delta\\!\psi)\cdots(J_{n}|\delta\\!\psi)\;\;\mathrm{d}\mathcal{R}_{*}\mu_{\psi_{0}}(\delta\\!\psi)\;\Bigg{/}\;\int\limits_{\delta\\!\psi\in\mathcal{R}[\mathbb{F}]}1\;\mathrm{d}\mathcal{R}_{*}\mu_{\psi_{0}}(\delta\\!\psi)$
(7)

with $\psi_{0}\in F$ and $J_{1},\dots,J_{n}\in\mathbb{F}^{*}$ as previously,
where $\mathcal{R}:\,\mathbb{F}\rightarrow\mathbb{F}$ is some continuous
linear operator, $\mathcal{R}[\mathbb{F}]\subset\mathbb{F}$ denotes the image
of $\mathbb{F}$ by $\mathcal{R}$, the symbol $\mu_{\psi_{0}}$ stands for the
pushforward of the hypothetical finite measure
$\rme^{\mathrm{i}\frac{1}{\hbar}S(\cdot)}\,\mathrm{d}\rho(\cdot)$ on $F$ via
the map $F\rightarrow\mathbb{F},\,\psi\mapsto(\psi{-}\psi_{0})$, and
$\mathcal{R}_{*}\mu_{\psi_{0}}$ stands for the pushforward of the measure
$\mu_{\psi_{0}}$ on $\mathbb{F}$ to $\mathcal{R}[\mathbb{F}]$ by
$\mathcal{R}$. The map $\mathcal{R}$ can be called a _regulator_ , and
typically it is a convolution operator by some test function in case of
theories over an affine spacetime (can be generalized for arbitrary spacetimes
as well), and Eq.(7) means nothing but the natural pushforward Feynman
integration on the subspace $\mathcal{R}[\mathbb{F}]\subset\mathbb{F}$, given
that the original Feynman integration Eq.(1) on $F$ was meaningful. The map
$\mathcal{R}$ implements the high frequency damping. Using the fundamental
formula of integral substitution, one infers that the Wilsonian regularized
MDS equation on the field correlators reads

$\displaystyle\Big{.}G^{(0)}=1\quad\mathrm{and}\quad\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi)}\,G=\mathrm{i}\,\hbar\,L_{\mathcal{R}\,\delta\\!\psi}\,G\qquad(\forall\,\delta\\!\psi\in\mathbb{F})$
(8)

in the analogy of Eq.(6), where again $L_{\mathcal{R}\,\delta\\!\psi}$ is the
left-multiplication in $\mathcal{T}(\mathbb{F})$ by the one-vector
$\mathcal{R}\,\delta\\!\psi\in\mathbb{F}$. As shall be expanded in Section 4,
the pertinent objects can be made well-defined similarly to that of the
unregularized MDS equation. The Wilsonian regularized MDS equation Eq.(8) does
not seem to be described in the literature.

From this point on, we drop the heuristic arguments, and all the statements
and formulas are intended to be mathematically rigorous. The aim of this paper
is to show that the objects involved in Eq.(6) and Eq.(8) are mathematically
well defined, and to establish the fundamental properties of the solution
spaces of the pertinent equations.

##  3 Mathematically rigorous definition of the unregularized MDS operator

As detailed in A, in a generic classical field theory, it is safe to assume
that the space of off-shell fields $F$ is the affine space of smooth sections
of a real finite dimensional affine bundle over a real finite dimensional
smooth base manifold. The space of field variations $\mathbb{F}$ are comprised
of differences of elements in $F$, and as such it is the vector space of
smooth sections of the real finite dimensional vector bundle subordinate to
our affine bundle, understood with the standard $\mathcal{E}$ smooth function
topology, which is known to be nuclear Fréchet. Within $\mathbb{F}$, there is
the space of test field variations $\mathbb{F}_{\\!{}_{T}}$, comprised of
compactly supported smooth sections, with the standard $\mathcal{D}$ test
function topology. For the sake of genericity, in this section we avoid using
the knowledge that $F$, $\mathbb{F}$ and $\mathbb{F}_{\\!{}_{T}}$ are these
concrete spaces, they will be considered abstract spaces instead. The symbol ∗
shall denote strong topological dual. See [22] and the Appendix of [11] for a
condensed summary on the theory of topological vector spaces.

######  Definition 1.

Let $F$ be a real affine space, with a subordinate real topological vector
space $\mathbb{F}$. Let the topology on $\mathbb{F}$ be nuclear Fréchet (in
short, NF space, see also [22]-RemarkLABEL:S-remNuclear, the $\mathcal{E}$
smooth function space is the archetype of an NF space). We call $F$ the space
of classical field configurations and the subordinate vector space
$\mathbb{F}$ the space of classical field variations. Let
$\mathbb{F}_{\\!{}_{T}}\subset\mathbb{F}$ be some subspace of $\mathbb{F}$,
endowed with a topology not weaker than $\mathbb{F}$. Let
$\mathbb{F}_{\\!{}_{T}}$ be either nuclear Fréchet or the strict inductive
limit of a countable system of nuclear Fréchet spaces with closed adjacent
images (in short, LNF space, see also [22]-RemarkLABEL:S-remNuclear, the
$\mathcal{D}$ test function space is the acrhetype of an LNF space). Then, we
call $\mathbb{F}_{\\!{}_{T}}$ the space of test field variations.

As detailed in A, in a generic concrete classical field theory, the
Euler–Lagrange functional is the derivative of the action functional, with its
linear variable restricted to the space of test field variations, so that the
Euler–Lagrange functional becomes an everywhere defined map. It is also shown
to be a jointly sequentially continuous map in its two variables. This
justifies the following abstract definition.

######  Definition 2.

Let
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R},\,(\psi,\delta\\!\psi_{\\!{}_{T}})\mapsto
E(\psi,\delta\\!\psi_{\\!{}_{T}})$ be a jointly sequentially continuous map
which is linear in its second variable. Then, $E$ will be called a classical
Euler–Lagrange (EL) functional. (By means of A Theorem45(ii,iii), then $E$ is
also separately continuous in its two variables, and when viewed as a map
$E:\,F\rightarrow\mathbb{F}_{\\!{}_{T}}^{*},\,\psi\mapsto E(\psi)$, it is
continuous.) Given a $\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$,
when the second argument of $E$ is evaluated, it will be denoted by
$(E\,|\,\delta\\!\psi_{\\!{}_{T}}):\,F\rightarrow\mathbb{R}$, which is then a
continuous map. When that map is evaluated at some $\psi\in F$, we denote it
by $\big{(}E(\psi)\,\big{|}\,\delta\\!\psi_{\\!{}_{T}}\big{)}\in\mathbb{R}$.
We call the equation

$\displaystyle\Big{.}\mathrm{we\;search\;for}\;\psi\in
F\;\mathrm{such\;that:}\qquad\forall\;\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\quad\big{(}E(\psi)\,\big{|}\,\delta\\!\psi_{\\!{}_{T}}\big{)}=0$
(9)

the classical Euler–Lagrange (EL) equation. When $E$ is viewed as a map
${E:\,F\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}}$, given any fixed field
$\psi_{0}\in F$, we use the notation
$E_{\psi_{0}}:=E\circ(\mathrm{I}_{\mathbb{F}}+\psi_{0})$, which will then be a
continuous map
$E_{\psi_{0}}:\,\mathbb{F}\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$, and
$\psi_{0}$ will be called a reference field. (By construction, for all
$\psi\in F$ the identity $E(\psi)=E_{\psi_{0}}(\psi{-}\psi_{0})$ holds.)

In order to define the MDS operator, we will need to invoke the notion of a
topologized tensor algebra made out of $\mathbb{F}$. For that, recall the
below facts.

######  Remark 3.

In this remark block let $\mathbb{U}$ denote a nuclear Fréchet (NF) or strong
dual of a nuclear Fréchet (DNF) space. (See also
[22]-RemarkLABEL:S-remNuclear.)

  1. (i)

For all $n\in\mathbb{N}_{0}$, the completed topological tensor product
$\mathop{\otimes}\limits^{n}\mathbb{U}$ is meaningful (e.g. understood with
the projective tensor product topology), and is NF or DNF, respectively.
Moreover, in the analogy of finite dimensional vector spaces, the pertinent
tensor product can be implemented via the multiplicative realization. That is,
it is topologically isomorphic to the space of the jointly continuous $n$-fold
multilinear forms on the strong dual space of $\mathbb{U}$. (See also
[22]-RemarkLABEL:S-remNuclear.)

  2. (ii)

With the same assumptions, one has that for all $n\in\mathbb{N}_{0}$, the
identity
$\big{(}\mathop{\otimes}\limits^{n}\mathbb{U}\big{)}^{*}\equiv\mathop{\otimes}\limits^{n}\mathbb{U}^{*}$
holds. (See also [22]-RemarkLABEL:S-remNuclear.)

  3. (iii)

Given a countable system of NF or a countable system of DNF spaces, their
cartesian product can be equipped with a vector space structure and with the
product (also called Tychonoff or initial or projective) topology. This is the
weakest topology such that the canonical projections of the cartesian product
are continuous. With this, it will become an NF or DNF space, respectively.
(See also [22]-RemarkLABEL:S-remCartesian, [22]-RemarkLABEL:S-remNuclear.)
Therefore, the Tychonoff tensor algebra
$\mathcal{T}(\mathbb{U}):=\mathop{\bigoplus}\limits_{n=0}^{\infty}\mathop{\otimes}\limits^{n}\mathbb{U}$
is meaningful and is NF or DNF, respectively. (The symbol
$\mathop{\bigoplus}\limits_{n=0}^{\infty}:=\mathop{\bigtimes}\limits_{n=0}^{\infty}$
as set operation, but we use rather $\bigoplus$ for vector spaces.)

  4. (iv)

Given a countable system of NF or a countable system of DNF spaces, in their
cartesian product vector space there is the subspace of the elements with all
zero except for finite entries, which subspace is called the algebraic direct
sum space. This can be equipped with the locally convex direct sum (also
called final or injective) topology. This is the strongest topology such that
the canonical injections of the cartesian product are continuous. With this,
it will become an NF or DNF space, respectively (see also
[22]-RemarkLABEL:S-remCartesian, [22]-RemarkLABEL:S-remNuclear). Therefore,
the algebraic tensor algebra
$\mathcal{T}_{a}(\mathbb{U}):=\mathop{\oplus}\limits_{n=0}^{\infty}\mathop{\otimes}\limits^{n}\mathbb{U}$
with the locally convex direct sum topology is meaningful and is NF or DNF,
respectively.

  5. (v)

One has that
$\big{(}\mathcal{T}(\mathbb{U})\big{)}^{*}\equiv\mathcal{T}_{a}(\mathbb{U}^{*})$
and
$\big{(}\mathcal{T}_{a}(\mathbb{U})\big{)}^{*}\equiv\mathcal{T}(\mathbb{U}^{*})$.
(See also [22]-RemarkLABEL:S-remCartesian.)

  6. (vi)

The Tychonoff tensor algebra has a jointly continuous bilinear map
${{\otimes}:\,{\mathcal{T}(\mathbb{U})\times\mathcal{T}(\mathbb{U})}\rightarrow\mathcal{T}(\mathbb{U})}$,
the tensor algebra multiplication, with a unit element
$\mathds{1}:=(1,0,0{\dots})\in\mathcal{T}(\mathbb{U})$ (consequences of
[22]-RemarkLABEL:S-remJoint). The subspaces of $k$-tensors provide a grading
of $\mathcal{T}(\mathbb{U})$. Quite trivially, the left multiplication
operator for all $u\in\mathcal{T}(\mathbb{U})$ is a continuous linear map
$L_{u}:\,\mathcal{T}(\mathbb{U})\rightarrow\mathcal{T}(\mathbb{U})$.

  7. (vii)

Similarly, the algebraic tensor algebra has a jointly continuous bilinear map
$\mathcal{T}_{a}(\mathbb{U})\times\mathcal{T}_{a}(\mathbb{U})\rightarrow\mathcal{T}_{a}(\mathbb{U})$,
the tensor algebra multiplication, with a corresponding unit element
(consequences of [22]-RemarkLABEL:S-remJoint). The subspaces of $k$-tensors
provide a grading of $\mathcal{T}_{a}(\mathbb{U})$. Quite trivially, the left
multiplication operator is a continuous linear map
$\mathcal{T}_{a}(\mathbb{U})\rightarrow\mathcal{T}_{a}(\mathbb{U})$.

  8. (viii)

Since $\mathcal{T}(\mathbb{U})$ and $\mathcal{T}_{a}(\mathbb{U}^{*})$ are
strong duals to each-other, and both of these are graded unital associative
algebras with jointly continuous multiplications, by transposing the algebra
multiplication and unit from the duals, one infers that both
$\mathcal{T}(\mathbb{U})$ and $\mathcal{T}_{a}(\mathbb{U}^{*})$ are
bialgebras, with corresponding coproduct and counit. The counit of
$\mathcal{T}(\mathbb{U})$ is
$b:\,\mathcal{T}(\mathbb{U})\rightarrow\mathbb{R},\,G:=(G^{(0)},G^{(1)},{\dots})\mapsto
b\,G:=G^{(0)}$, i.e. extraction of the scalar component, the symbol “$b$”
standing for “base” or “bottom form”.

  9. (ix)

Due to the bialgebra nature of $\mathcal{T}(\mathbb{U})$, i.e. due to the
existence of a continuous coproduct on $\mathcal{T}(\mathbb{U})$, for all
$p\in\mathcal{T}_{a}(\mathbb{U}^{*})$ the corresponding left insertion
operator
$\text{\Large$\iota$}_{p}:\,\mathcal{T}(\mathbb{U})\rightarrow\mathcal{T}(\mathbb{U})$
is meaningful, and is a continuous linear operator. More concretely, the left
insertion operator $\text{\Large$\iota$}_{p^{(n)}}$ with a
$p^{(n)}\in\mathop{\otimes}\limits^{n}\mathbb{U}^{*}$ ($n\in\mathbb{N}_{0}$)
exists, because for all $m\in\mathbb{N}_{0}$ ($m\geq n$) the tensor product
$\mathop{\otimes}\limits^{m}\mathbb{U}$ can be identified with the space of
$\mathop{\otimes}\limits^{n}\mathbb{U}^{*}\;\times\;\mathop{\otimes}\limits^{m{-}n}\mathbb{U}^{*}\rightarrow\mathbb{R}$
jointly continuous bilinear forms, as stated in (i). Similarly, the left
insertion operators make sense in $\mathcal{T}_{a}(\mathbb{U})$, and is a
continuous linear operator. (For the sake of distinction in terminology, we
call merely the operators $\text{\Large$\iota$}_{p^{(1)}}$ with
$p^{(1)}\in\mathbb{U}^{*}\equiv\mathop{\otimes}\limits^{1}\mathbb{U}^{*}\subset\mathcal{T}_{a}(\mathbb{U}^{*})$
as insertion operators, whereas for generic
$p^{(n)}\in\mathop{\otimes}\limits^{n}\mathbb{U}^{*}\subset\mathcal{T}_{a}(\mathbb{U}^{*})$
($n\in\mathbb{N}_{0}$) or more generally for
$p\in\mathcal{T}_{a}(\mathbb{U}^{*})$, we call the corresponding
$\text{\Large$\iota$}_{p^{(n)}}$ or $\text{\Large$\iota$}_{p}$ as
multipolynomial insertion operator.) For all
$p\in\mathcal{T}_{a}(\mathbb{U}^{*})$, one has the identity
$p=b\,\text{\Large$\iota$}_{p}$. For the left insertion operator, we use the
normalization convention such that for all
$G^{(n)}\in\mathop{\otimes}\limits^{n}\mathbb{U}$ and $u\in\mathbb{U}$ and
$p\in\mathbb{U}^{*}$ one has
$\text{\Large$\iota$}_{p}\,L_{u}\;G^{(n)}=(n+1)\,(p|u)\,G^{(n)}$.

  10. (x)

A historical note: over an affine (Minkowski) spacetime, one can define the
space of rapidly decreasing (Schwartz) functions $\mathcal{S}$, which is an NF
space. The tensor algebra $\mathcal{T}_{a}(\mathcal{S})$ is referred to as
Borchers–Uhlmann (BU) algebra (original papers: [23, 24], and including a
short review: [25]). The Wightman functionals in QFT are understood to be in
the space
$\big{(}\mathcal{T}_{a}(\mathcal{S})\big{)}^{*}\equiv\mathcal{T}(\mathcal{S}^{*})$.

  11. (xi)

By construction, the $\mathcal{T}_{a}$ topology is strongest tensor algebra
topology, whereas $\mathcal{T}$ is the weakest. It is possible to define a
natural topological tensor algebra which is in between the $\mathcal{T}$ and
$\mathcal{T}_{a}$, in terms of topology strength. It will be motivated and
introduced later, in Section 5, and will be key to the presented construction,
if one wishes to quantize analytic EL functionals, and not only polynomial
ones.

######  Definition 4.

Assume that the space of field variations as real nuclear Fréchet spaces has a
direct sum splitting $\mathbb{F}=\mathbb{F}_{r}\oplus\mathbb{F}_{c}$, called
the real-complex splitting, where both $\mathbb{F}_{r}$ and $\mathbb{F}_{c}$
are closed (and therefore nuclear Fréchet), and $\mathbb{F}_{c}$ has a complex
structure (i.e. it can be regarded as a complex nuclear Fréchet space). Denote
by $\mathbb{F}_{r\mathbb{C}}:=\mathbb{F}_{r}\otimes\mathbb{C}$ the
complexification of $\mathbb{F}_{r}$. Then, we use the notation
$\mathbb{F}_{(\mathbb{C})}:=\mathbb{F}_{r\mathbb{C}}\oplus\mathbb{F}_{c}$, and
call it the space of field variations with complex structure. (We assume that
also $\mathbb{F}_{\\!{}_{T}}\subset\mathbb{F}$ respects this splitting.)

The above definition is necessary, because in field theory, certain fields
(like variations of Dirac fields) sit in an inherently complex vector space,
whereas other fields (like variations of gauge fields) sit in an inherently
real vector space, and QFT assumes that the sectors not being inherently
complex are complexified. In the most simple case, one has merely
$\mathbb{F}_{(\mathbb{C})}=\mathbb{F}$ if $\mathbb{F}$ was complex, or
$\mathbb{F}_{(\mathbb{C})}=\mathbb{F}\otimes\mathbb{C}$ if $\mathbb{F}$ was
real.

######  Definition 5.

Let the vector space of field variations admit a real-complex splitting
$\mathbb{F}={\mathbb{F}_{r}\oplus\mathbb{F}_{c}}$, as in Definition4.
Furthermore, assume a direct sum structure
$\mathbb{F}=\mathop{\oplus}\limits_{i=1}^{f}\mathbb{F}_{i}$, such that for
each $i=1,{\dots},f$ the subspace $\mathbb{F}_{i}$ is either entirely within
$\mathbb{F}_{r}$ or in $\mathbb{F}_{c}$ and are closed (thus, also nuclear
Fréchet), and let there be integers $s_{i}\in\\{0,1\\}$ associated to each
subspace $\mathbb{F}_{i}$ ($i=1,{\dots},f$). Then, the subspaces
$\mathbb{F}_{1},{\dots},\mathbb{F}_{f}$ are called the flavor sectors, and
their associated integers $s_{1},{\dots},s_{f}$ are called bosonic or
fermionic labels. (We assume that also
$\mathbb{F}_{\\!{}_{T}}\subset\mathbb{F}$ respects this splitting.)

In the most simple case, there is only one single flavor sector, globally
endowed with a bosonic or fermionic label. For invoking the MDS equation, we
will need the graded-symmetrized subspace of
$\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$, according to the bosonic and
fermionic labels. In order to establish that algebra, the following remark is
useful.

######  Remark 6.

Whenever $\mathbb{F}$ is split as
$\mathbb{F}=\mathop{\oplus}\limits_{i=1}^{f}\mathbb{F}_{i}$ into flavor
sectors with bosonic / fermionic labels $s_{i}$ ($i=1,{\dots},f$), as in
Definition5, then for all $n\in\mathbb{N}_{0}$ one may introduce a continuous
linear representation $U_{\pi}$ of a permutation group element $\pi\in\Pi_{n}$
on the space $\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$ as follows
(see also [26] Chapter4). Take an element
$x_{1}{\otimes}{\dots}{\otimes}x_{n}\in\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$,
where each factor $x_{i}$ ($i=1,{\dots},n$) resides in some
$\mathbb{F}_{(\mathbb{C})j}$ ($j=1,{\dots},f$). Then, set

$\displaystyle U_{\pi}(x_{1}{\otimes}{\dots}{\otimes}x_{n})$ $\displaystyle:=$
$\displaystyle(-1)^{s_{1}\,\sigma_{1}(\pi)+{\dots}+s_{f}\,\sigma_{f}(\pi)}\;x_{\pi(1)}{\otimes}{\dots}{\otimes}x_{\pi(n)},$

where $\sigma_{i}(\pi)\in\\{0,1\\}$ ($i=1,{\dots},f$) is the parity of the
perumation $\pi$ within each index block. The map $U_{\pi}$ can then be
linearly extended in $\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$.
Due to the NF property of the involved spaces, the topology defining seminorms
on $\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$ may be taken to be
such that $U_{\pi}$ are continuous ([26] Chapter4), therefore can uniquely be
extended as acting as a continuous linear map
$U_{\pi}:\,\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}\rightarrow\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$,
thus defining the signed permutation operator on the entire space
$\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$. Therefore, on each
space $\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$ the continuous
linear projection operator

$\displaystyle P_{n}:=\frac{1}{n!}\;\sum_{\pi\in\Pi_{n}}U_{\pi}$

can be defined. The family of operators $P_{n}$ ($n\in\mathbb{N}_{0}$) on the
spaces $\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$ can be joined as
a single grading preserving continuous linear projection operator
${P:\,\mathcal{T}(\mathbb{F}_{(\mathbb{C})})\rightarrow\mathcal{T}(\mathbb{F}_{(\mathbb{C})})}$.
This signed symmetrizer projection operator $P$ has the following properties
against the tensor algebra multiplication:

$\displaystyle
P(x\,y)\;=\;P(P(x)\,y)\;=\;P(x\,P(y))\;=\;P(P(x)\,P(y))\quad(\forall\,x,y\in\mathcal{T}(\mathbb{F}_{(\mathbb{C})})).$

Therefore, the closed subspace $\mathop{\mathrm{Ker}}\nolimits(P)$ is a two-
sided ideal in $\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$. (The presented
approach was inspired by [26] Chapter4.)

Using the fact that the closed subspace of an NF space is also NF and that the
factor space of an NF space with a closed subspace is also NF (see also
[22]-RemarkLABEL:S-remNuclear), the following definition is meaningful.

######  Definition 7.

Let the space of field variations $\mathbb{F}$ admit flavor sectors
$\mathbb{F}_{i}$ and bosonic / fermionic labels $s_{i}$ ($i=1,{\dots},f$), as
in Definition5, and corresponding signed symmetrization projector $P$ as in
Remark6. Then the factor algebra
$A(\mathbb{F}_{(\mathbb{C})}):=\mathcal{T}(\mathbb{F}_{(\mathbb{C})})/\mathop{\mathrm{Ker}}\nolimits(P)$
is called the field algebra. Clearly, it is a unital associative algebra, and
a nuclear Fréchet (NF) topological vector space, with jointly continuous
algebra multiplication $\bullet:\,A(\mathbb{F}_{(\mathbb{C})})\times
A(\mathbb{F}_{(\mathbb{C})})\rightarrow A(\mathbb{F}_{(\mathbb{C})})$. The
topological transpose $P^{*}$ of $P$ allows the analogous construction in the
strong dual of $\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$, which makes it also a
unital associative algebra with jointly continuous algebra multiplication, and
therefore $A(\mathbb{F}_{(\mathbb{C})})$ retains the bialgebra structure from
$\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$.

Since the complementing projection operator $I{-}P$ to $P$ is also continuous,
as topological vector spaces one may naturally identify
$A(\mathbb{F}_{(\mathbb{C})})$ with the closed subspace
$\mathop{\mathrm{Ker}}\nolimits(I{-}P)=\mathop{\mathrm{Ran}}\nolimits(P)\subset\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$.
Using this linear topological identification, the algebraic product $\bullet$
may be pushed forward from $A(\mathbb{F}_{(\mathbb{C})})$ to the subspace
$\mathop{\mathrm{Ran}}\nolimits(P)\subset\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$.
That is, as usual, the algebra $A(\mathbb{F}_{(\mathbb{C})})$ may be regarded
as a closed subspace of $\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$. On that
space the product $\bullet$ can be traced back to the tensor algebra product
$\otimes$, with the identity: for all
$x\in\mathop{\otimes}\limits^{m}\mathbb{F}_{(\mathbb{C})}$ and
$y\in\mathop{\otimes}\limits^{n}\mathbb{F}_{(\mathbb{C})}$, one has $x\bullet
y=\frac{(m+n)!}{m!\,n!}\,P(x\otimes y)$. The unit element, the counit map, as
well as the insertion operator by a one-form
$p^{(1)}\in\mathbb{F}_{(\mathbb{C})}^{*}$ coincides to the one defined on
$\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$. The strong dual of
$A(\mathbb{F}_{(\mathbb{C})})$ may be identified with the corresponding
subspace of $\mathcal{T}_{a}(\mathbb{F}_{(\mathbb{C})}^{*})$. Whenever not
confusing, we will suppress the multiplication symbol $\bullet$.

The above definition was necessary, because in QFT the Feynman type field
correlators are graded-symmetrized, i.e. they sit rather in
$A(\mathbb{F}_{(\mathbb{C})})$ than in
$\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$. (In the most simple case one has
that $A(\mathbb{F}_{(\mathbb{C})})$ is $\bigvee(\mathbb{F}_{(\mathbb{C})})$ or
$\bigwedge(\mathbb{F}_{(\mathbb{C})})$.) As expanded above, the left
multiplication operator (given some $\delta\\!\psi\in\mathbb{F}$) is the same
as $L_{\delta\\!\psi}$ in $\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$, with a
subsequent graded-symmetrization and combinatorial normalization. It shall be
denoted by the same symbol $L_{\delta\\!\psi}$, when not confusing. According
to the chosen normalization conventions, in the algebra
$A(\mathbb{F}_{(\mathbb{C})})\subset\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$,
the counit map $b$ and the left-insertion operator $\text{\Large$\iota$}_{p}$
by a one-form $p\in\mathbb{F}_{(\mathbb{C})}^{*}$ literally coincide with the
corresponding operators in $\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$. Due to
the graded-symmetrization, one has that for all $G\in
A(\mathbb{F}_{(\mathbb{C})})$, and for all
$\delta\\!\psi\in\mathbb{F}_{(\mathbb{C})}$,
$\delta\\!J\in\mathbb{F}_{(\mathbb{C})}^{*}$ from the same fermionic sector
$\left(\text{\Large$\iota$}_{\delta\\!J}\,L_{\delta\\!\psi}+L_{\delta\\!\psi}\,\text{\Large$\iota$}_{\delta\\!J}\right)\,G=(\delta\\!J|\delta\\!\psi)\,G$
holds, whereas
$\left(\text{\Large$\iota$}_{\delta\\!J}\,L_{\delta\\!\psi}-L_{\delta\\!\psi}\,\text{\Large$\iota$}_{\delta\\!J}\right)\,G=(\delta\\!J|\delta\\!\psi)\,G$
holds otherwise.

Until Section 5, for the sake of simplicity we assume that the EL functional
$E:\,F\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$ is multipolynomial, which is
defined as follows.

######  Definition 8.

Let $E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R}$ be an EL
functional as in Definition2. We say that the EL functional $E$ is
multipolynomial, whenever there exists a reference field $\psi_{0}\in F$, such
that there exists an element
$\mathbf{E}_{\psi_{0}}\in\big{(}A(\mathbb{F}_{(\mathbb{C})})\big{)}^{*}\otimes\mathbb{F}_{\\!{}_{T}}^{*}\subset\mathcal{T}_{a}(\mathbb{F}_{(\mathbb{C})}^{*})\otimes\mathbb{F}_{\\!{}_{T}}^{*}$,
for which

$\displaystyle\forall\,\psi\in
F,\;\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\quad\big{(}E_{\psi_{0}}(\psi{-}\psi_{0})\,\big{|}\,\delta\\!\psi_{\\!{}_{T}}\big{)}=\Big{(}\mathbf{E}_{\psi_{0}}\,\Big{|}\,\big{(}1,\,\mathop{\otimes}\limits^{1}(\psi{-}\psi_{0}),\,\mathop{\otimes}\limits^{2}(\psi{-}\psi_{0}),\,\dots\big{)}\otimes\delta\\!\psi_{\\!{}_{T}}\Big{)}$

holds. (Note that then for all $\psi_{0}\in F$ there exists the corresponding
element $\mathbf{E}_{\psi_{0}}$.) When an element
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$ is contracted with
$\mathbf{E}_{\psi_{0}}$ in its last tensorial entry, we will use the notation
$(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})$ to denote the
corresponding element of
$\big{(}A(\mathbb{F}_{(\mathbb{C})})\big{)}^{*}\subset\mathcal{T}_{a}(\mathbb{F}_{(\mathbb{C})}^{*})$.

Given $(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})$ as above, it has a
corresponding multipolynomial insertion operator over the tensor algebra
$\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$, as stated in Remark3(ix). We shall
denote that by the symbol
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}$.

######  Definition 9.

Let $\hbar$ be a fixed real number. Let $F$, $\mathbb{F}$,
$\mathbb{F}_{\\!{}_{T}}$ as in Definition1. Let
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R}$ as in Definition2,
and assume that it is multipolynomial as in Definition8. Let
$A(\mathbb{F}_{(\mathbb{C})})$ be the field algebra as in Definition7. Then,
for some fixed reference field $\psi_{0}\in F$ and fixed test field
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$ the operator

$\displaystyle\mathbf{M}_{\hbar,\psi_{0},\delta\\!\psi_{\\!{}_{T}}}:\quad
A(\mathbb{F}_{(\mathbb{C})})\rightarrow A(\mathbb{F}_{(\mathbb{C})}),\quad
G\mapsto\mathbf{M}_{\hbar,\psi_{0},\delta\\!\psi_{\\!{}_{T}}}\,G\,:=\,\Big{(}\,\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}\;-\;\mathrm{i}\,\hbar\,L_{\delta\\!\psi_{\\!{}_{T}}}\,\Big{)}\,G$
(10)

is called the unregularized master Dyson–Schwinger (MDS) operator. We call the
below equation the unregularized master Dyson–Schwinger (MDS) equation:

$\displaystyle\Big{.}\mathrm{we\;search\;for}\;(\psi_{0},G_{\psi_{0}})\in
F\times A(\mathbb{F}_{(\mathbb{C})}),\;\mathrm{such\;that:}$ (11)
$\displaystyle\Big{.}b\,G_{\psi_{0}}=1,\qquad\mathrm{and}\qquad\forall\;\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\quad\mathbf{M}_{\hbar,\psi_{0},\delta\\!\psi_{\\!{}_{T}}}\,G_{\psi_{0}}=0.$
(12)

The MDS formulation of QFT can be though of as a construction, where the
objects of interest are elements of $F\times A(\mathbb{F}_{(\mathbb{C})})$,
and the selection equation for the physically realized such elements is the
MDS equation. In Section 4 it shall be shown that some finetuning
(regularization) to this idea is needed, as is well known in the QFT
literature.

######  Definition 10.

Any continuous map $O:\,F\rightarrow\mathbb{R}$ is called an observable,
similarly as in a classical field theory. Given a fixed $\psi_{0}\in F$, we
use the notation $O_{\psi_{0}}:=O\circ(\mathrm{I}_{\mathbb{F}}+\psi_{0})$,
which is then a continuous map
$O_{\psi_{0}}:\,\mathbb{F}\rightarrow\mathbb{R}$, and one has
$O(\psi)=O_{\psi_{0}}(\psi{-}\psi_{0})$ for all $\psi\in F$ and observable
$O$. An observable $O:\,F\rightarrow\mathbb{R}$ is called multipolynomial
observable, whenever for some reference fields $\psi_{0}\in F$, there exists
an element $\mathbf{O}_{\psi_{0}}\in\mathcal{T}_{a}(\mathbb{F}^{*})$, such
that for all $\psi\in F$, one has
$O_{\psi_{0}}(\psi{-}\psi_{0})=\Big{(}\mathbf{O}_{\psi_{0}}\,\Big{|}\,\big{(}1,\,\mathop{\otimes}\limits^{1}(\psi{-}\psi_{0}),\,\mathop{\otimes}\limits^{2}(\psi{-}\psi_{0}),\,\dots\big{)}\Big{)}$.
(If it holds, it then holds for any $\psi_{0}\in F$.)

######  Definition 11.

Given a solution $(\psi_{0},G_{\psi_{0}})\in F\times
A(\mathbb{F}_{(\mathbb{C})})$ of the MDS equation, the (Feynman type) quantum
expectation value of the multipolynomial observable
$O:\,F\rightarrow\mathbb{R}$ at the solution $(\psi_{0},G_{\psi_{0}})$ is
$\mu_{{}_{(\psi_{0},G_{\psi_{0}})}}(O):=\left(\mathbf{O}_{\psi_{0}}\,\big{|}\,G_{\psi_{0}}\right)$.

We note that the above construction can be extended also to non-polynomial but
analytic EL functionals and observables as well. For that, however, a stronger
topology is needed on the tensor algebra of $\mathbb{F}$, which we will
address later in Section 5.

######  Example 12.

For a scalar $\varphi^{4}$ model over a fixed Minkowski spacetime
$\mathcal{M}$, the MDS operator reads as follows. Let $\mathrm{v}$ be the
affine constant maximal form over $\mathcal{M}$ (corresponding to the Lebesgue
measure). Denote by $\Box$ the Minkowski wave operator. Set
$F:=\mathbb{F}:=C^{\infty}(\mathcal{M},\mathbb{R})$ and
$\mathbb{F}_{\\!{}_{T}}:=C^{\infty}_{c}(\mathcal{M},\mathbb{R})$. Then, the EL
functional is
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R},\,(\psi,\,\delta\\!\psi_{\\!{}_{T}})\mapsto\int_{\mathcal{M}}\delta\\!\psi_{\\!{}_{T}}\,\Box\psi\,\mathrm{v}+\int_{\mathcal{M}}\delta\\!\psi_{\\!{}_{T}}\,\psi^{3}\,\mathrm{v}$.
For any fixed test field
${\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}}$ the corresponding MDS
operator can be expressed as

$\displaystyle\big{(}\mathbf{M}_{\hbar,\psi_{0},\delta\\!\psi_{\\!{}_{T}}}\,G\big{)}^{(n)}(x_{1},{\dots},x_{n})=$
(13)
$\displaystyle\quad\int_{y\in\mathcal{M}}\delta\\!\psi_{\\!{}_{T}}(y)\,\Box_{y}G^{(n+1)}(y,x_{1},{\dots},x_{n})\,\mathrm{v}(y)+\int_{y\in\mathcal{M}}\delta\\!\psi_{\\!{}_{T}}(y)\,G^{(n+3)}(y,y,y,x_{1},{\dots},x_{n})\,\mathrm{v}(y)$
(14)
$\displaystyle\qquad\Big{.}-\mathrm{i}\,\hbar\,n\,\tfrac{1}{n!}\textstyle\sum_{\pi\in\Pi_{n}}\delta\\!\psi_{\\!{}_{T}}(x_{\pi(1)})\,G^{(n-1)}(x_{\pi(2)},{\dots},x_{\pi(n)})$
(15)

at the reference field $\psi_{0}=0$ (for all $G\in
A(\mathbb{F}_{(\mathbb{C})})\,{=}\bigvee(\mathbb{F}\otimes\mathbb{C})$ and
$n\in\mathbb{N}_{0}$ and $x_{1},{\dots},x_{n}\in\mathcal{M}$, where $\Pi_{n}$
denotes the set of permutations of the symbols $1,{\dots},n$).

######  Example 13.

For a pure Yang–Mills model (possibly non-abelian, i.e. self-interacting) over
a fixed spacetime $(\mathcal{M},g_{ab})$, the MDS operator reads as follows
(Penrose abstract indices abc… and abc… are used for tangent tensors and their
duals, respectively). Let $\mathrm{v}$ be the canonical volume form associated
to the spacetime metric $g_{ab}$. Let $F$ denote the affine space of covariant
derivation operators over some vector bundle $V(\mathcal{M})$ with some given
structure group $\mathcal{G}$ (internal or gauge group). Then, any two
covariant derivations $\nabla,\nabla^{\prime}\in F$ has a difference tensor
(Yang–Mills potential) $A:=\nabla^{\prime}-\nabla$ residing in the space of
smooth sections of
$T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})$,
denoted by $\mathbb{F}$. Let $\mathbb{F}_{\\!{}_{T}}$ denote the space
compactly supported sections from $\mathbb{F}$. Then, the EL functional is
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R},\,(\nabla,\,A_{T})\mapsto\int_{\mathcal{M}}A_{T}{}_{d}\cdot\left(-\tilde{\nabla}_{e}\,\mathrm{v}\,g^{ec}g^{db}P(\nabla)_{cb}\right)$,
the symbol $P(\nabla)_{ab}$ denoting the curvature tensor of $\nabla$ and
$\cdot$ denoting the pointwise trace form on the sections of
$V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})$, whereas $\tilde{\nabla}$ denoting
an extension of $\nabla$ to the mixed tensor algebra of $V(\mathcal{M})$ and
$T(\mathcal{M})$ with an arbitrary torsion-free covariant derivation on
$T(\mathcal{M})$. (The pertinent differential operator expression involving
$\tilde{\nabla}$ is known to be uniquely defined, see also Remark 42 in A.)
Specially, fix a covariant derivation $\nabla\in F$ as a reference field, then
the EL functional with respect to this reference field $\nabla$ reads as
$E_{\nabla}:\,\mathbb{F}\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R},\,(A,\,A_{T})\mapsto\int_{\mathcal{M}}A_{T}{}_{d}\cdot\left(-\tilde{\nabla}_{e}\,\mathrm{v}\,g^{ec}g^{db}\,(\tilde{\nabla}_{c}A_{b}-\tilde{\nabla}_{b}A_{c}+[A_{c},A_{b}])\right)$.
Given a test field variation $A_{T}\in\mathbb{F}_{\\!{}_{T}}$, the
corresponding MDS operator is

$\displaystyle\big{(}\mathbf{M}_{\hbar,\nabla,A_{T}}\,G\big{)}^{(n)}(x_{1},{\dots},x_{n})_{a_{1}{\dots}a_{n}}{}^{\alpha_{1}{\dots}\alpha_{n}}=$
(16)
$\displaystyle\quad\int_{y\in\mathcal{M}}\Big{(}-A_{T}(y)_{d}{}^{\gamma}\,K(y)_{\gamma\beta}\,\tilde{\nabla}^{y}_{e}\,\mathrm{v}(y)\,g(y)^{ec}\,g(y)^{db}\,\tilde{\nabla}^{y}_{c}\,G^{(n+1)}(y,x_{1},{\dots},x_{n})_{ba_{1}{\dots}a_{n}}{}^{\beta\alpha_{1}{\dots}\alpha_{n}}$
(17)
$\displaystyle\Big{.}\qquad\quad\;\;+A_{T}(y)_{d}{}^{\gamma}\,K(y)_{\gamma\beta}\,\tilde{\nabla}^{y}_{e}\,\mathrm{v}(y)\,g(y)^{ec}\,g(y)^{db}\,\tilde{\nabla}^{y}_{b}\,G^{(n+1)}(y,x_{1},{\dots},x_{n})_{ca_{1}{\dots}a_{n}}{}^{\beta\alpha_{1}{\dots}\alpha_{n}}$
(18) $\displaystyle\Big{.}\quad-
A_{T}(y)_{d}{}^{\gamma}\,K(y)_{\gamma\beta}\,\tilde{\nabla}^{y}_{e}\,\mathrm{v}(y)\,g(y)^{ec}\,g(y)^{db}\,G^{(n+2)}(y,y,x_{1},{\dots},x_{n})_{[cb]a_{1}{\dots}a_{n}}{}^{\delta\,\varepsilon\,\alpha_{1}{\dots}\alpha_{n}}\,C(y)_{\delta\,\varepsilon}^{\beta}\Big{)}$
(19)
$\displaystyle\quad\Big{.}-\mathrm{i}\,\hbar\,n\,\tfrac{1}{n!}\textstyle\sum_{\pi\in\Pi_{n}}A_{T}(x_{\pi(1)})_{a_{\pi(1)}}{}^{\alpha_{\pi(1)}}\,G^{(n-1)}(x_{\pi(2)},{\dots},x_{\pi(n)})_{a_{\pi(2)}{\dots}a_{\pi(n)}}{}^{\alpha_{\pi(2)}{\dots}\alpha_{\pi(n)}}$
(20)

for all $G\in
A(\mathbb{F}_{(\mathbb{C})})\,{=}\bigvee(\mathbb{F}\otimes\mathbb{C})$ and
$n\in\mathbb{N}_{0}$ and $x_{1},{\dots},x_{n}\in\mathcal{M}$, where $\Pi_{n}$
denotes the set of permutations of the symbols $1,{\dots},n$, and the Penrose
abstract indices ${}^{\alpha\,\beta\,\gamma\,\delta\,\varepsilon\dots}$ were
used for the Lie algebra of the structure group $\mathcal{G}$, with
$K_{\alpha\beta}$ denoting the index notation of the trace form, and
$C_{\alpha\beta}^{\gamma}$ denoting the index notation of the commutator.

##  4 The weak (distributional) and the Wilsonian regularized MDS operator

######  Definition 14.

Let
$F,\,\mathbb{F},\,\mathbb{F}_{\\!{}_{T}},\,A(\mathbb{F}_{(\mathbb{C})}),\,E$
be as in Definition9. We call the EL functional
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R}$ to be free or non-
interacting, whenever the corresponding continuous map
$E:\,F\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$ is affine. We call the
Euler–Lagrangian functional interacting otherwise. (Note that by construction,
for a free EL functional, given any reference field $\psi_{0}\in F$, the map
$E_{\psi_{0}}(\cdot)-E_{\psi_{0}}(0):\,\mathbb{F}\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$
is linear. If in addition, $\psi_{0}$ were an EL solution, then
$E_{\psi_{0}}(\cdot):\,\mathbb{F}\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$ is
linear.)

######  Remark 15.

It is seen that if $(\psi_{0},\,G_{\psi_{0}})\in F\times
A(\mathbb{F}_{(\mathbb{C})})$ were a solution to the unregularized MDS
equation Eq.(12), and the reference field $\psi_{0}\in F$ is an EL solution,
and $E$ is non-interacting, then
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}\,G_{\psi_{0}}^{(2)}=\mathrm{i}\,\hbar\,\delta\\!\psi_{\\!{}_{T}}$
holds for all test fields
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$.

######  Corollary 16.

Let the EL functional $E$ be the one of the free wave or Klein–Gordon equation
over Minkowski spacetime. In that case, the solution space of the
unregularized MDS equation Eq.(12) is empty, whenever $\hbar\neq 0$.

The above is rather evident by means of Remark15: the correlator
$G_{\psi_{0}}^{(2)}$ would need to be proportional to a fundamental solution
(Green’s functional), which does not sit in the space of smooth correlators
$\mathbb{F}{\otimes}\mathbb{F}$, but is at best a distribution. It is thus
tempting to extend the definition of the MDS equation in the weak
(distributional) sense, so that free theories can have MDS solutions. In order
to define the distributional sense fields, one needs to use the information
that the EL functional
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R}$ is actually that of a
concrete classical field theory. Namely, that $F$ is the space of smooth
sections of an affine bundle, $\mathbb{F}$ is the space of smooth sections of
its subordinate vector bundle, and $\mathbb{F}_{\\!{}_{T}}$ is the space of
compactly supported smooth sections of that vector bundle.

######  Remark 17.

In order to define the weak MDS operator, we will need to substitute
$A(\mathbb{F}_{(\mathbb{C})})$ with its distributional version, which is
expanded below.

  1. (i)

Assume that $F$ is the space of smooth sections of an affine bundle over the
base manifold $\mathcal{M}$, with subordinate vector bundle $U(\mathcal{M})$,
whose smooth sections span the space $\mathbb{F}$. Take the densitised dual of
that vector bundle,
${U\\!^{\times}\\!(\mathcal{M}):=U^{*}(\mathcal{M})\otimes\mathop{\wedge}\limits^{\dim(\mathcal{M})}T^{*}(\mathcal{M})}$,
and denote the space of its smooth sections by $\mathbb{F}^{\times}$
correspondingly. Take the $n$-fold external tensor product bundle
${U\\!^{\times}\\!(\mathcal{M})\boxtimes\dots\boxtimes
U\\!^{\times}\\!(\mathcal{M})}$ of that, which will then be a vector bundle
over the $n$-fold cartesian product
${\mathcal{M}\times\dots\times\mathcal{M}}$ as base manifold. The space of
smooth sections of this vector bundle shall be denoted by
$\mathbb{F}^{\times}_{n}$, which has its natural $\mathcal{E}$ topology which
is nuclear Fréchet (NF), and is topologically isomorphic to
$\mathop{\otimes}\limits^{n}\mathbb{F}^{\times}$ by means of Schwartz kernel
theorem. It has the subspace of compactly supported sections, denoted by
$\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}$ and is a dense subspace within
$\mathbb{F}^{\times}_{n}$ in the $\mathcal{E}$ topology. The space
$\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}$ with its natural $\mathcal{D}$
topology becomes a countable strict inductive limit of nuclear Fréchet spaces
with closed adjacent images (LNF space) whenever the base manifold
$\mathcal{M}$ is noncompact, and is nuclear Fréchet (NF) if $\mathcal{M}$ is
compact (see [22]-RemarkLABEL:S-remNuclear). The strong dual of the space
$\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}$ is denoted by
$(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}\big{)}^{*}$ with its natural
$\mathcal{D}^{*}$ topology. It is a DLNF space when $\mathcal{M}$ is
noncompact, and $DNF$ when $\mathcal{M}$ is compact. One has that
$\mathop{\otimes}\limits^{n}\mathbb{F}\subset(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}\big{)}^{*}$,
i.e. the latter space can be regarded as the space of distributional $n$-field
correlators.

  2. (ii)

In the above construction we avoided using completed topological tensor
product $\mathop{\otimes}\limits^{n}\mathbb{F}_{\\!{}_{T}}^{\times}$, as that
space is topologically not isomorphic to
$\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}$ whenever we are in the realm of LNF
spaces, i.e. when $\mathcal{M}$ is noncompact (although they are isomorphic as
linear spaces, the latter has a stronger topology). This slight complication
is mentioned in more details in
[22]-RemarkLABEL:S-remJoint(LABEL:S-remJointDD). The pertinent issue is
absent, whenever $\mathcal{M}$ is compact
($\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}\equiv\mathop{\otimes}\limits^{n}\mathbb{F}_{\\!{}_{T}}^{\times}$
topologically, in that case, i.e. one does not need to distinguish them on
compact manifolds).

  3. (iii)

One can form the algebraic tensor algebra
$\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$, defined as the algebraic
direct sum
$\mathop{\oplus}\limits_{n=0}^{\infty}\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n}$
equipped with the locally convex direct sum topology. Its topology will be LNF
whenever $\mathcal{M}$ is noncompact, and NF if $\mathcal{M}$ is compact.
$\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$ forms a unital associative
algebra, with (at least) separately continuous multiplication.

  4. (iv)

The tensor algebra of distributional field variations
$\mathcal{T}\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times})^{*}\big{)}$ is defined to
be the space
$\big{(}\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\big{)}^{*}$. It is
topologically isomorphic to
$\mathop{\bigoplus}\limits_{n=0}^{\infty}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{n})^{*}$,
by means of [22]-RemarkLABEL:S-remCartesian. It is a DLNF space when
$\mathcal{M}$ is noncompact, and DNF space if $\mathcal{M}$ is compact. It is
also a unital associative algebra, with an (at least) separately continuous
algebra multiplication.

  5. (v)

The distributional graded-symmetrized field algebra
$A\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}\big{)}\subset\mathcal{T}\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}\big{)}$
can be defined in the analogy of Definition7. Clearly, the smooth field
algebra $A(\mathbb{F}_{(\mathbb{C})})$, is dense in the distributional sense
field algebra
$A\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}\big{)}$.

######  Remark 18.

The MDS operator of a non-interacting EL functional can be naturally extended
in the distributional sense, as follows.

  1. (i)

A continuous linear operator $A:\,\mathbb{F}\rightarrow\mathbb{F}$ is said to
possess a formal transpose, if there exsists a continuous linear operator
$A^{t}:\,\mathbb{F}_{\\!{}_{T}}^{\times}\rightarrow\mathbb{F}_{\\!{}_{T}}^{\times}$,
such that for all $\delta\\!\psi\in\mathbb{F}$ and
$p_{{}_{T}}\in\mathbb{F}_{\\!{}_{T}}^{\times}$ one has that
$\int_{\mathcal{M}}(A\,\delta\\!\psi)\,p_{{}_{T}}=\int_{\mathcal{M}}\delta\\!\psi\,(A^{t}\,p_{{}_{T}})$,
with $\mathcal{M}$ being the underlying manifold. The topological transpose
${\big{(}A^{t}\big{)}^{*}:\,\big{(}\mathbb{F}_{\\!{}_{T}}^{\times}\big{)}^{*}\rightarrow\big{(}\mathbb{F}_{\\!{}_{T}}^{\times}\big{)}^{*}}$
of the formal transpose operator is called the distributional extension of
$A$.

  2. (ii)

The notion of formal transpose can be generalized to operators
$A:\,\mathcal{T}(\mathbb{F})\rightarrow\mathcal{T}(\mathbb{F})$, being of the
type
$A^{t}:\,\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\rightarrow\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$,
and
$\big{(}A^{t}\big{)}^{*}:\,\big{(}\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\big{)}^{*}\rightarrow\big{(}\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\big{)}^{*}$
being the distributional extension of $A$.

  3. (iii)

One may note that for all $\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$
and $G\in A(\mathbb{F}_{(\mathbb{C})})$ and
$p\in\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$ one has that
$(p\,|\,L_{\delta\\!\psi_{\\!{}_{T}}}G)=(\text{\Large$\iota$}_{\delta\\!\psi_{\\!{}_{T}}}p\,|\,G)$.
Moreover, the linear map
$\text{\Large$\iota$}_{\delta\\!\psi_{\\!{}_{T}}}:\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\rightarrow\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$
is continuous. Therefore, $\text{\Large$\iota$}_{\delta\\!\psi_{\\!{}_{T}}}$
is the formal transpose of $L_{\delta\\!\psi_{\\!{}_{T}}}$. Consequently, the
operator $L_{\delta\\!\psi_{\\!{}_{T}}}$ admits a distributional extension
$A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})\rightarrow
A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})$, being the
topological transpose of $\text{\Large$\iota$}_{\delta\\!\psi_{\\!{}_{T}}}$.

  4. (iv)

Whenever $E$ is the EL functional of a non-interacting classical field theory,
and $\psi_{0}\in F$ is fixed, then for each
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$ there exists a unique
element $\pi_{{}_{T}}\in\mathbb{F}_{\\!{}_{T}}^{\times}$, such that
$(E_{\psi_{0}}(\delta\\!\psi)|\delta\\!\psi_{\\!{}_{T}})-(E(\psi_{0})|\delta\\!\psi_{\\!{}_{T}})=\int_{\mathcal{M}}\pi_{{}_{T}}\,\delta\\!\psi$
for all $\delta\\!\psi\in\mathbb{F}$ (see also A). Therefore, one has that
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})-(E(\psi_{0})|\delta\\!\psi_{\\!{}_{T}})}=\text{\Large$\iota$}_{\int_{\mathcal{M}}\pi_{{}_{T}}\,(\cdot)}$.
Because of that, for any $G\in A(\mathbb{F}_{(\mathbb{C})})$ and
$p\in\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$ one has the identity
$\big{(}p\,\big{|}\,\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})-(E(\psi_{0})|\delta\\!\psi_{\\!{}_{T}})}\,G\big{)}=\big{(}L_{\pi_{{}_{T}}}\,p\,\big{|}\,G\big{)}$,
i.e. the formal transpose of
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}$
exists, being the continuous linear map
$L_{\pi_{{}_{T}}}+(E(\psi_{0})|\delta\\!\psi_{\\!{}_{T}})\,I:\,\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\rightarrow\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$.
Consequently, the operator
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}$
admits a distributional extension
$A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})\rightarrow
A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})$.

  5. (v)

The above construction clearly fails for interacting classical field theories,
since then the formal transpose of
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}$ as
a continuous linear map
$\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})\rightarrow\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$
cannot be defined. See e.g. the interaction term in Eq.(15) as an example.

  6. (vi)

Let $E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R}$ be the EL
functional of a classical field theory, and $J\in\mathbb{F}_{\\!{}_{T}}^{*}$,
then we call an element $\mathtt{K}_{J}\in F$ a solution with a source $J$
whenever
$\forall\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\;(E(\mathtt{K}_{J})\,|\,\delta\\!\psi_{\\!{}_{T}})=(J|\delta\\!\psi_{\\!{}_{T}})$
holds. Specially, one may consider only
$J\in\mathbb{F}_{\\!{}_{T}}^{\times}\subset\mathbb{F}^{\times}\subset\mathbb{F}_{\\!{}_{T}}^{*}$.
If $\mathtt{K}:\,\mathbb{F}_{\\!{}_{T}}^{\times}\rightarrow F$ is a continous
map, such that for all $J\in\mathbb{F}_{\\!{}_{T}}^{\times}$ the field
$\mathtt{K}(J)\in F$ is a solution with a source $J$, then $\mathtt{K}$ is
called a fundamental solution. (It may or may not exist, and if exists, it is
typically not unique.)

######  Definition 19.

Let
$F,\,\mathbb{F},\,\mathbb{F}_{\\!{}_{T}},\,A(\mathbb{F}_{(\mathbb{C})}),\,E,\,\hbar$
be as in Definition9, and let
$E:\,F\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{R}$ the EL functional of
a non-interacting classical field theory as in the Definition14. Fix a
reference field $\psi_{0}\in F$. Then, by means of Remark18, for all
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$, the MDS operator can be
extended as a continuous linear operator
${\mathbf{M}_{\hbar,\psi_{0},\delta\\!\psi_{\\!{}_{T}}}:\,A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})\rightarrow
A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})}$, called to be the
weak or distributional master Dyson–Schwinger (MDS) operator. We call the
equation

$\displaystyle\Big{.}\mathrm{we\;search\;for}\;G_{\psi_{0}}\in
A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}),\;\mathrm{\;such\;that:}$
(21)
$\displaystyle\Big{.}b\,G_{\psi_{0}}=1,\qquad\mathrm{and}\qquad\forall\;\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\quad\mathbf{M}_{\hbar,\psi_{0},\delta\\!\psi_{\\!{}_{T}}}\,G_{\psi_{0}}=0.$
(22)

the weak or distributional master Dyson–Schwinger (MDS) equation.

######  Remark 20.

With the above notations, assume that the EL equation admits a fundamental
solution $\mathtt{K}:\,\mathbb{F}_{\\!{}_{T}}^{\times}\rightarrow
F,\,J\mapsto\mathtt{K}(J)$ as in Remark18(vi). Then,
$E_{\psi_{0}}:\,\mathbb{F}\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$ also has a
corresponding fundamental solution
$\mathtt{K}_{\psi_{0}}:\,\mathbb{F}_{\\!{}_{T}}^{\times}\rightarrow\mathbb{F},\,J\mapsto\mathtt{K}_{\psi_{0}}(J):=\mathtt{K}(J){-}\psi_{0}$.
Let $\psi_{0}$ be an EL solution, in which case
$E_{\psi_{0}}:\,\mathbb{F}\rightarrow\mathbb{F}_{\\!{}_{T}}^{*}$ becomes
linear, and assume that $\mathtt{K}_{\psi_{0}}$ can be chosen to be linear.
Such a linear fundamental solution
$\mathtt{K}_{\psi_{0}}:\,\mathbb{F}_{\\!{}_{T}}^{\times}\rightarrow\mathbb{F}$
can be naturally considered as an element
$\mathtt{K}_{\psi_{0}}^{(2)}\in\mathcal{L}(\mathbb{F}_{\\!{}_{T}}^{\times},\mathbb{F})\subset(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{2})^{*}$.
Assume moreover, that $\mathtt{K}_{\psi_{0}}^{(2)}$ can be chosen to be
invariant to the permutation symmetry of the field algebra. (E.g. for a wave
or Klein–Gordon equation over Minkowski spacetime, the Feynman propagator
would be such.)

  1. (i)

Given these conditions, one may define the element
$K_{\psi_{0}}:=(0,\,0,\,\mathrm{i}\,\hbar\,\mathtt{K}_{\psi_{0}}^{(2)},\,0,\,0,{\dots})\in
A\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}\big{)}$, called
to be the connected correlator, and one can take the ansatz
$G_{\psi_{0}}:=\exp(K_{\psi_{0}})\in
A\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}\big{)}$. Then,
$(\psi_{0},G_{\psi_{0}})\in F\times
A\big{(}(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*}\big{)}$ solves
the weak (distributional sense) MDS equation Eq.(22).

  2. (ii)

For the bosonic case, the above statement is seen trivially, by the fact that
for all $\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$ the insertion
operator
$\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}$ is
an algebra derivation, and the field algebra is commutative, so one can use
the formula for the derivative of exponential. If $\mathbb{F}$ has fermionic
flavor sectors as well, then one can still trace the problem back to
derivations acting on exponential: whenever
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$ resides in a single
flavor sector, then for all $\delta\\!\psi_{\\!{}_{T}}^{\prime}$ from the same
flavor sector, the linear map
$L_{\delta\\!\psi_{\\!{}_{T}}^{\prime}}\,\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}$
is also an algebra derivation.

  3. (iii)

Rather evidently, the above do not necessarily exhaust all the possible
solutions. Typically, a fundamental solution $\mathtt{K}_{\psi_{0}}$
satisfying the above is not unique. Moreover, one may add any term
$\delta\\!K_{\psi_{0}}$ to $K_{\psi_{0}}$ satisfying
$b\,\delta\\!K_{\psi_{0}}=0$ and
$\forall\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\;\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}\,\delta\\!K_{\psi_{0}}=0$,
in which case $\exp(K_{\psi_{0}}+\delta\\!K_{\psi_{0}})$ will still solve the
weak MDS equation. In usual QFTs, these ambiquities are removed by further
invariance requirements on $G_{\psi_{0}}$, which are not dealt with in the
present paper.

  4. (iv)

The existence of the assumed type of fundamental solution is guaranteed for
any EL functional over an affine base manifold, whenever $E_{\psi_{0}}$
corresponds to a linear PDE with a multipolynomial differential operator,
having constant coefficients. This is ensured by the celebrated
Malgrange–Ehrenpreis theorem ([27] ChapitreI.1 Theoréme1 and [28] Chapter6
Theorem10).

######  Corollary 21.

Let the EL functional $E$ be the one of a non-interacting classical field
theory, which has a fundamental solution as in Remark20. In that case, the
solution space of the weak (distributional) MDS equation Eq.(22) is not empty.

According to Remark20, given a reference field $\psi_{0}\in F$, the solutions
of the weak MDS equation Eq.(22) are not unique for free EL functionals. In
the usual QFT constructions, this ambiquity is removed by additional
requirements, such as Poincaré invariance of the solutions. In the presented
construction, however such auxiliary conditions are not imposed, since in a
generally covariant setting, it is not evident that the vacuum state should be
required to be unique or not.

######  Remark 22.

For interacting models, the followings can be stated.

  1. (i)

On one hand, there is a negative result: for a generic non-interacting EL
functional, the unregularized MDS equation Eq.(12) has no solutions.

  2. (ii)

On the other hand, there is a positive result: for a generic non-interacting
EL functional of a classical field theory having appropriate fundamental
solution, the weak MDS equation Eq.(22) does have solutions, just as is the
common wisdom in heuristic QFT.

  3. (iii)

For interacting models, the weak MDS operator cannot be an everywhere defined
continuous operator acting on the space of distributional correlators. For
instance, in a $\varphi^{4}$ model over Minkowski spacetime, the interacting
part of the MDS operator does not have a formal transpose, as seen in Eq.(15).
This phenomenon occurs because the diagonal evaluation map of smooth functions
${\big{(}(x,y)\mapsto G(x,y)\big{)}\longmapsto\big{(}z\mapsto G(z,z)\big{)}}$
$x,y,z\in\mathcal{M}$ cannot be extended to the distributions, in general.

  4. (iv)

In order to remedy the above problem, one is tempted to view the everywhere
defined continuous bilinear operator
$\mathbf{M}_{\hbar,\psi_{0}}:\,A(\mathbb{F}_{(\mathbb{C})})\times\mathbb{F}_{\\!{}_{T}}\rightarrow
A(\mathbb{F}_{(\mathbb{C})})$ as a densely defined bilinear operator
$\hat{\mathbf{M}}_{\hbar,\psi_{0}}:\,A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})\times\mathbb{F}_{\\!{}_{T}}\rightarrowtail
A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})$, via the natural
dense linear embedding $A(\mathbb{F}_{(\mathbb{C})})\subset
A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})$ of the function
sense correlators to the distributional sense correlators. Then, one is
tempted to take its maximally extended operator, understood by its sequential
closure. That is, a distributonal correlator would be in the domain of the
extended $\hat{\mathbf{M}}_{\hbar,\psi_{0}}$, whenever it admits a function
sense approximating sequence converging to it in the distributional sense,
such that the evaluated $\mathbf{M}_{\hbar,\psi_{0}}$ on the approximator
sequence is convergent in the distributional sense. The operator would be
closable, whenever any two such approximator sequence of the same domain
element yielded the same result. This strategy is made impossible by the fact
that for all interacting EL functionals one can show that the MDS operator is
not sequentially closable. (This occurs because the above diagonal evaluation
map is so-called maximally non-closable, see [22]-RemarkLABEL:S-remClosure for
more details.)

  5. (v)

The celebrated Hörmander’s criterion [29] on the wave front set gives a
sufficient condition for diagonal evaluation of multivariate distributions,
but that condition is not applicable for the present problem. (E.g. already
the wave front set of a solution to the distributional MDS equation generated
from the Minkowski wave or Klein–Gordon equation is known to fail Hörmander’s
sufficiency criterion, see [30] Chapter 4 and [31].)

  6. (vi)

One can prove that the solution space of the unregularized MDS equation
(understood over the smooth correlators) is always empty, regardless of the
structure of the underlying base manifold $\mathcal{M}$ and the interactions
in the EL functional (we plan to detail the proof in a different paper).

In summary, the problem is that for interacting models, only the function
sense MDS operator is well defined, but its solution space is always empty. In
the non-interacting case, the MDS operator can be extended in the
distributional sense, and its solution space has the right properties.
However, the distributional extension of the MDS operator cannot be achieved
for interacting models. In order to overcome this difficulty, one needs the
regularized MDS operator, introduced below.

######  Definition 23.

Let
$F,\,\mathbb{F},\,\mathbb{F}_{\\!{}_{T}},\,A(\mathbb{F}_{(\mathbb{C})}),\,E,\,\hbar$
be as in Definition9. Fix a continuous linear operator
$\mathcal{R}:\,\mathbb{F}\rightarrow\mathbb{F}$. Given these, we call the
operator

$\displaystyle\mathbf{M}_{\hbar,\psi_{0},\mathcal{R},\delta\\!\psi_{\\!{}_{T}}}:\;A(\mathbb{F}_{(\mathbb{C})})\rightarrow
A(\mathbb{F}_{(\mathbb{C})}),\;G\mapsto\mathbf{M}_{\hbar,\psi_{0},\mathcal{R},\delta\\!\psi_{\\!{}_{T}}}\,G:=\Big{(}\,\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}-\mathrm{i}\,\hbar\,L_{\mathcal{R}\,\delta\\!\psi_{\\!{}_{T}}}\,\Big{)}\,G$
(23)

the $\mathcal{R}$-regularized master Dyson–Schwinger (MDS) operator. Moreover,
we call

$\displaystyle\Big{.}\mathrm{we\;search\;for}\;(\psi_{0},G_{\psi_{0}})\in
F\times A(\mathbb{F}_{(\mathbb{C})}),\;\mathrm{such\;that:}$ (24)
$\displaystyle\Big{.}b\,G_{\psi_{0}}=1,\quad\mathrm{and}\quad\forall\;\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\;\mathbf{M}_{\hbar,\psi_{0},\mathcal{R},\delta\\!\psi_{\\!{}_{T}}}\,G_{\psi_{0}}=0$
(25)

the $\mathcal{R}$-regularized master Dyson–Schwinger (MDS) equation.

The above definition is motivated by the Wilsonian regularization,
heuristically stated in Eq.(8). If the base manifold $\mathcal{M}$ were an
affine space, in order to achieve a Wilsonian regularization (UV frequency
damping), the regularizer operator $\mathcal{R}$ should be chosen as the
convolution operator by a test function on $\mathcal{M}$. It is not difficult
to see that in such a setting, for the non-interacting case, the Wilsonian
regularized MDS equation Eq.(25) does have solutions in the space of smooth
field correlators. Thus, Definition23 is expected to make sense also for
interacting theories, since there is no problem with the diagonal evaluation
map on the space of multivariate smooth functions. In order to adapt this
construction to generic, non-affine manifolds $\mathcal{M}$, we invoke a
notion of generalized convolution on smooth manifolds, see also [32, 33, 34].

######  Remark 24.

The following terminologies are standard in the theory of pseudodifferential
operators [32, 33, 34], and generalizes the notion of convolution to
manifolds.

  1. (i)

A continuous linear map
$C:\,{(\mathbb{F}^{\times})^{*}{\rightarrow}\mathbb{F}}$ is called a smoothing
operator, their space is denoted by $\Psi^{-\infty}$ in the literature. By
Schwartz kernel theorem, such an operator can be identified with an element
$\kappa\in\mathbb{F}{\otimes}\mathbb{F}^{\times}$, i.e. $\kappa$ is a smooth
section of the vector bundle $U(\mathcal{M})\boxtimes
U^{{\times}}(\mathcal{M})$ over the base $\mathcal{M}{\times}\mathcal{M}$,
with $\mathbb{F}$ being the space of smooth sections of $U(\mathcal{M})$. This
is emphasized by writing $C_{\kappa}$ instead, where
$\big{(}C_{\kappa}\,\delta\\!\psi_{\\!{}_{T}}\big{)}(x)=\int_{y\in\mathcal{M}}\kappa(x,y)\,\delta\\!\psi_{\\!{}_{T}}(y)$
for all
$\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}\subset(\mathbb{F}^{\times})^{*}$
and for all $x\in\mathcal{M}$.

  2. (ii)

A smoothing operator $C_{\kappa}$ is called properly supported, whenever the
canonical projections from
$\mathop{\mathrm{supp}}\nolimits(\kappa)\subset\mathcal{M}{\times}\mathcal{M}$
onto each factor $\mathcal{M}$ is proper, i.e. the inverse images of compact
sets are compact. In other words, for all compact subsets
$\mathcal{K}\subset\mathcal{M}$ the closure of the sets
${\\{(x,y)\in\mathcal{M}{\times}\mathcal{M}\,|\,x\in\mathcal{K},\;\kappa(x,y)\neq
0\\}}$ and
${\\{(x,y)\in\mathcal{M}{\times}\mathcal{M}\,|\,y\in\mathcal{K},\;\kappa(x,y)\neq
0\\}}$ are compact. In that case, the map $C_{\kappa}$ can act as continuous
linear maps $\mathbb{F}_{\\!{}_{T}}\rightarrow\mathbb{F}_{\\!{}_{T}}$,
$\mathbb{F}\rightarrow\mathbb{F}$,
$(\mathbb{F}^{\times})^{*}\rightarrow(\mathbb{F}^{\times})^{*}$,
$(\mathbb{F}_{\\!{}_{T}}^{\times})^{*}\rightarrow(\mathbb{F}_{\\!{}_{T}}^{\times})^{*}$.

######  Definition 25.

Let $C_{\kappa}$ be a properly supported smoothing operator as in Remark24. If
it preserves the flavor sectors, then $\kappa$ is called a mollifying kernel.

######  Remark 26.

A special example can shed some light on the role of $\kappa$ in Definition25.
Let $\mathcal{M}$ be a finite dimensional real affine space (“Minkowski
spacetime”), the subordinate finite dimensional real vector space denoted by
$T$ (“tangent space”). Let the vector bundle of fields be trivial, and
trivialized compatibly with the affine structure. In that case, the fields,
i.e. the elements of $\mathbb{F}$ are simply smooth functions from
$\mathcal{M}$ to a finite dimensional real vector space, in which the
classical fields take their values. Let us denote the identity operator of
that finite dimensional real vector space by $I$. Due to the affineness of
$\mathcal{M}$, up to a positive multiplier there exists a unique positive
volume form field $\mathrm{v}$ which is parallel against the affine parallel
transport (this corresponds to the Lebesgue measure). Take a compactly
supported $C^{\infty}$ real valued scalar field
$\rho:\,T\rightarrow\mathbb{R}$. Then, the field
$(x,y)\mapsto\kappa(x,y):=\rho(x{-}y)\,\mathrm{v}(y)\,I$ is called a
convolution kernel, and defines a mollifying kernel. For any element
$\delta\\!\psi\in\mathbb{F}$ one has then that
$C_{\kappa}\,\delta\\!\psi=\rho\star\delta\\!\psi$, i.e. $C_{\kappa}$ is the
convolution operator by $\rho$. Similarly, for any element
$p\in\mathbb{F}^{\times}$ one has that $C_{\kappa}^{t}\,p=\rho^{t}\star p$,
where $\rho^{t}$ is the reflected $\rho$ (for all $v\in T$,
$\rho^{t}(v):=\rho({-}v)$). Due to the compact support of $\rho$, the $\kappa$
is indeed properly supported. Moreover, by construction, it is flavor sector
preserving.

With the notion of mollifying kernel, one can define the Wilsonian
regularization (UV frequency cutoff) also over generic manifolds. Namely, in
Definition23, one sets $\mathcal{R}=C_{\kappa}$ for some mollifying kernel
$\kappa$. In that case, we use the abbreviation
$\mathbf{M}_{\hbar,\psi_{0},\kappa,\delta\\!\psi_{\\!{}_{T}}}$ for
$\mathbf{M}_{\hbar,\psi_{0},\mathcal{R},\delta\\!\psi_{\\!{}_{T}}}=\mathbf{M}_{\hbar,\psi_{0},C_{\kappa},\delta\\!\psi_{\\!{}_{T}}}$.
It is seen that the regularized MDS equation is the analogy of the
unregularized MDS equation Eq.(12), but with a smoothing appearing in it.

######  Remark 27.

The following observation is useful for constructing concrete solutions of the
regularized MDS equation. Let $\mathcal{R}:\,\mathbb{F}\rightarrow\mathbb{F}$
be a continuous linear operator (typically, a smoothing operator $C_{\kappa}$
in our example). Then, it can be uniquely extended as a continuous grading
preserving algebra derivation
$\mathcal{R}:\,\mathcal{T}(\mathbb{F})\rightarrow\mathcal{T}(\mathbb{F})$ of
the unital associative topological graded algebra $\mathcal{T}(\mathbb{F})$
via requiring the annihilation of unity ($\mathcal{R}\,\mathds{1}=0$), the
preservation of the space of $n$-tensors ($n\in\mathbb{N}_{0}$), the Leibniz
rule over tensor product, and coincidence with $\mathcal{R}$ on the one-
vectors. If $\mathcal{R}$ is also preserving flavor sectors of $\mathbb{F}$,
then it can be restricted to
$A(\mathbb{F}_{(\mathbb{C})})\subset\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$ as
an algebra derivation. Similarly, the topological transpose operator
$\mathcal{R}^{*}:\,\mathbb{F}^{*}\rightarrow\mathbb{F}^{*}$ extends as a
continuous linear operator
$\mathcal{R}^{*}:\,\mathcal{T}_{a}(\mathbb{F}^{*})\rightarrow\mathcal{T}_{a}(\mathbb{F}^{*})$.
Assume moreover, that the pertinent operator $\mathcal{R}$ on $\mathbb{F}$ has
a formal transpose
$\mathcal{R}^{t}:\,\mathbb{F}_{\\!{}_{T}}^{\times}\rightarrow\mathbb{F}_{\\!{}_{T}}^{\times}$.
Then, for the same reason it extends uniquely to
$\mathcal{T}_{a}(\mathbb{F}_{\\!{}_{T}}^{\times})$ in the above manner, and as
$(\mathcal{R}^{t})^{*}$ to
$\mathcal{T}((\mathbb{F}_{\\!{}_{T}}^{\times})^{*})$, and thus also to
$A((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})\subset\mathcal{T}((\mathbb{F}_{\\!{}_{T}}^{\times}{}_{(\mathbb{C})})^{*})$,
if $\mathcal{R}$ was flavor sector preserving. The operator
$(\mathcal{R}^{t})^{*}$ will not be distinguised in notation from
$\mathcal{R}$, since the former is the distributional extension of the latter.
Similarly, $\mathcal{R}^{t}$ will in general be denoted by $\mathcal{R}^{*}$,
since the latter is the distributional extension of the former.

######  Remark 28.

Use the assumptions of Remark20, and let $\kappa$ be a mollifying kernel. In
that case, the $\kappa$-regularized fundamental solution
$\tfrac{1}{2}C_{\kappa}\,\mathtt{K}_{\psi_{0}}^{(2)}$ resides in
$\mathop{\otimes}\limits^{2}\mathbb{F}$, and it is compatible with the
permutation symmetry of the field algebra $A(\mathbb{F}_{(\mathbb{C})})$. Let
the base manifold be affine, and let $\kappa$ be specifically a convolution
kernel by a symmetric test function. Then,
$\tfrac{1}{2}C_{\kappa}\,\mathtt{K}_{\psi_{0}}^{(2)}$ satisfies
$\forall\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}:\;\text{\Large$\iota$}_{(\mathbf{E}_{\psi_{0}}|\delta\\!\psi_{\\!{}_{T}})}\,\tfrac{1}{2}C_{\kappa}\,\mathtt{K}_{\psi_{0}}^{(2)}=L_{C_{\kappa}\delta\\!\psi_{\\!{}_{T}}}\,\mathds{1}$.
Define the element
$K_{\psi_{0},\kappa}:=(0,\,0,\,\mathrm{i}\,\hbar\,\tfrac{1}{2}C_{\kappa}\,\mathtt{K}_{\psi_{0}}^{(2)},\,0,\,0,{\dots})\in
A(\mathbb{F}_{(\mathbb{C})})$, called to be the smoothed connected correlator.
Define the smoothed correlator with the ansatz
$G_{\psi_{0},\kappa}:=\exp(K_{\psi_{0},\kappa})\in
A(\mathbb{F}_{(\mathbb{C})})$. Then, $(\psi_{0},G_{\psi_{0},\kappa})\in
F\times A(\mathbb{F}_{(\mathbb{C})})$ solves the $\kappa$-regularized MDS
equation Eq.(25). In order to see this, one merely needs to repeat the proof
of Remark20. (The object $\tfrac{1}{2}C_{\kappa}\,\mathtt{K}_{\psi_{0}}^{(2)}$
e.g. for a wave or Klein–Gordon model over Minkowski spacetime, would
correspond to the smoothed Feynman propagator.)

It is seen that in the above definition the trick is that although the
fundamental solution
$\mathtt{K}_{\psi_{0}}^{(2)}\in\mathcal{L}(\mathbb{F}_{\\!{}_{T}}^{\times},\mathbb{F})\subset(\mathbb{F}_{\\!{}_{T}}^{\times}{}_{2})^{*}$
is merely defined in the distributional sense, but its $\kappa$-regularized
version $\tfrac{1}{2}C_{\kappa}\,\mathtt{K}_{\psi_{0}}^{(2)}$ sits in the
space of smooth field correlators $\mathbb{F}{\otimes}\mathbb{F}$. Therefore,
free theories of such kind will have smooth solutions of the
$\kappa$-regularized MDS equation, and one does not need to go to the realm of
distributional sense MDS equation, which is not applicable to interacting
models. The replacement of the unregularized MDS operator with the
$\kappa$-regularized MDS operator is called _regularization_. A further sanity
check on the presented Wilsonian regularization scheme of the MDS equation is
the fact that such an equation would always have formal perturbative solutions
if the field algebra were $\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$, and the
base manifold $\mathcal{M}$ were affine. This can be seen to be an immediate
consequence of the Malgrange–Ehrenpreis surjectivity theorem ([27] ChapitreI.1
Theoréme1 and [28] Chapter6 Theorem10), but will be expanded in a different
paper.

Having a rigorous and generally covariant formulation of Wilsonian
regularization at hand, it is natural to ask the question whether it is
possible to formulate Wilsonian renormalization using that. The answer is
affirmative, and is addressed in B.

##  5 An existence condition for regularized MDS solutions

In this section, we present an existence condition for the solutions of the
Wilsonian regularized MDS equation.

######  Remark 29.

The followings spell out some facts about the topology of $\mathbb{F}$ and
$\mathbb{F}_{\\!{}_{T}}$.

  1. (i)

Assume that the base manifold $\mathcal{M}$ underlying a concrete classical
field theory is compact (with or without boundary, and if with boundary, we
assume the cone condition). This is a realistic assumption for conformally
invariant models, as for those, the theory can be reformulated on the compact
manifold with boundary, underlying the conformally compactified spacetime.

  2. (ii)

With the assumption as above, rather obviously, the space
$\mathbb{F}_{\\!{}_{T}}$ shall become also an NF space, similarly to
$\mathbb{F}$. That is, $\mathbb{F}_{\\!{}_{T}}$ becomes metrizable with all of
its benefits: its topology will be sequential, and separetely continuous
multilinear maps from it will become jointly continuous. (Recall that if
$\mathcal{M}$ is compact then, either $\mathbb{F}_{\\!{}_{T}}=\mathbb{F}$, or
$\mathbb{F}_{\\!{}_{T}}$ may be chosen to be the closed subspace of
$\mathbb{F}$ consisting of fields vanishing at $\partial\mathcal{M}$ together
with all of their derivatives. Its natural topology will become an
$\mathcal{E}$ function topology instead of $\mathcal{D}$ type.)

  3. (iii)

It is also an elementary fact that over a compact base manifold $\mathcal{M}$,
the space $\mathbb{F}$ and $\mathbb{F}_{\\!{}_{T}}$ will not only be nuclear
Fréchet, but also will admit continuous norms instead of merely continuous
seminorms. By means of [22]-RemarkLABEL:S-remCH, then they become nuclear
Fréchet spaces with a countable increasing system of topology defining
Hilbertian norms. Since these norms are simply Sobolev norms, they are
Gel’fand compatible (see [22]-RemarkLABEL:S-remCH and
[22]-RemarkLABEL:S-remSobolev), therefore by means of [22]-RemarkLABEL:S-remCH
they become NF spaces with the countably Hilbert (CH) property. Recall that if
$\mathcal{H}$ is a CH type NF space, then there exists a countable family

$\displaystyle H_{0}\supset H_{1}\supset{\dots}\supset H_{n}\supset{\dots}$

of topological vector spaces, such that
$\mathcal{H}=\mathop{\bigcap}\limits_{n\in\mathbb{N}_{0}}H_{n}$ and is dense
in all of the spaces $H_{n}$ ($n\in\mathbb{N}_{0}$), moreover their topologies
are gradually strictly strengthening

$\displaystyle\tau_{0}|_{{}_{\mathcal{H}}}\subset\tau_{1}|_{{}_{\mathcal{H}}}\subset{\dots}\subset\tau_{n}|_{{}_{\mathcal{H}}}\subset{\dots},$

and all of their topologies are complete and generated by a Hilbertian scalar
product (that is, for each $n\in\mathbb{N}_{0}$, the space $H_{n}$ can be
taken to be a Hilbert space), and for all $n\in\mathbb{N}_{0}$ there exists an
integer $m\geq 1$ such that the inclusion maps $i_{n+m,n}:\,H_{n+m}\rightarrow
H_{n}$ are nuclear. (Specially, the spaces $H_{0},\,H_{1},\,{\dots}$ may be
chosen such that all adjacent inclusion maps are nuclear.) The respective
topology generating Hilbertian norms thus form an increasing system

$\displaystyle\|\cdot\|_{0}\leq C_{0,1}\,\|\cdot\|_{1}\leq{\dots}\leq
C_{n{-}1,n}\,\|\cdot\|_{n}\leq{\dots}$ $\displaystyle\qquad\Big{.}(\text{for
some }C_{0,1},{\dots},C_{n-1,n},{\dots}\in\mathbb{R}^{+}).$

The corresponding Hilbertian scalar products shall be denoted by

$\displaystyle\left<\cdot,\cdot\right>_{0},\;\left<\cdot,\cdot\right>_{1},\;{\dots},\;\left<\cdot,\cdot\right>_{n},\;{\dots}.$

The proposed existence theorem will hinge on the fact that on the field
algebra $A(\mathbb{F}_{(\mathbb{C})})$ it is possible to naturally define a
reasonable topology, somewhat stronger than the Tychonoff topology, such that
it preserves the NF property coming from $\mathbb{F}$, and if present, the
eventual CH property of $\mathbb{F}$ as well.

######  Remark 30.

We recall some findings on topologies of the tensor algebra of $\mathbb{F}$
[35].

  1. (i)

Dubin and Hennings in their work ([35] Chapter3.1) introduces the notion of
tensor algebra topology of the following kind. Let $\mathbb{F}$ be a nuclear
Fréchet space. Then, they define the vector space
$\mathcal{T}(\mathbb{F},\lambda,\mathfrak{p})$, where $\lambda$ is some
topological subspace of the space of $\mathbb{N}_{0}\rightarrow\mathbb{C}$
sequences (it is a so-called _Köthe echelon space_), and $\mathfrak{p}$ is a
family of Hilbertian seminorms on $\mathbb{F}$ defining its NF topology
(recall that multiple seminorm families $\mathfrak{p}$ can define the same
topology on $\mathbb{F}$). As a vector space, it is defined as follows:

$\displaystyle\mathcal{T}(\mathbb{F},\lambda,\mathfrak{p})$ $\displaystyle:=$
$\displaystyle\Big{\\{}G\in\mathop{\bigoplus}\limits_{n=0}^{\infty}\mathop{\otimes}\limits^{n}\mathbb{F}\;\Big{|}\;\forall\|{\cdot}|\in\mathfrak{p}:\,\big{(}\|G^{(n)}|^{{\otimes}n}\big{)}_{n\in\mathbb{N}_{0}}\in\lambda\Big{\\}},$

where for all topology defining Hilbertian seminorms
$\|{\cdot}|\in\mathfrak{p}$ on $\mathbb{F}$, the symbol
$\|{\cdot}|^{{\otimes}n}$ denotes the $n$-fold cross norm over
$\mathop{\otimes}\limits^{n}\mathbb{F}$ originating from $\|{\cdot}|$, which
is then also a Hilbertian seminorm ([35] Chapter3.1). The locally convex
vector topology on $\mathcal{T}(\mathbb{F},\lambda,\mathfrak{p})$ is defined
by the system of seminorms
$\big{(}G\mapsto\sum_{n=0}^{\infty}|u_{n}|\;\|G^{(n)}|^{{\otimes}n}\big{)}_{u\in\lambda\\!^{\times},\|{\cdot}|\in\mathfrak{p}}$,
where $\lambda\\!^{\times}$ denotes the so-called Köthe dual (which is, under,
mild conditions, the strong topological dual) of the sequence space $\lambda$.

  2. (ii)

Notable Köthe echelon spaces include ([35] Chapter2.4):

$\displaystyle\phi\,$ $\displaystyle:=$
$\displaystyle\\{(u_{n})_{n\in\mathbb{N}_{0}}\in\mathbb{C}^{\mathbb{N}_{0}}\,|\,\exists
m\in\mathbb{N}_{0}:\,\forall n>m:\,u_{n}=0\\}\quad\text{(terminating
sequences)},$ $\displaystyle h\,$ $\displaystyle:=$
$\displaystyle\\{(u_{n})_{n\in\mathbb{N}_{0}}\in\mathbb{C}^{\mathbb{N}_{0}}\,|\,\forall
m\in\mathbb{N}_{0}:\,\exists C_{m}\in\mathbb{R}^{+}:\,\forall
n\in\mathbb{N}_{0}:\,2^{n\,m}|u_{n}|\leq C_{m}\\},$ $\displaystyle h^{\prime}$
$\displaystyle:=$
$\displaystyle\\{(u_{n})_{n\in\mathbb{N}_{0}}\in\mathbb{C}^{\mathbb{N}_{0}}\,|\,\exists
c\in\mathbb{N}:\,\exists m_{c}\in\mathbb{N}_{0}:\,\forall
n\in\mathbb{N}_{0}:\,|u_{n}|\leq c\,2^{n\,m_{c}}\\},$ $\displaystyle\omega\,$
$\displaystyle:=$
$\displaystyle\\{(u_{n})_{n\in\mathbb{N}_{0}}\in\mathbb{C}^{\mathbb{N}_{0}}\,|\,\text{any
}u\\}\qquad\qquad\qquad\qquad\qquad\text{(all sequences)}.$

These are understood with their so-called normal topologies ([35] Chapter2.1).
The space $\omega$ is the space of all sequences with the natural Tychonoff
topology, the space $\phi$ is the space of finitely terminating sequences with
the natural locally convex direct sum topology, whereas the space $h$ is known
to be topologically isomorphic to the space $H(\mathbb{C})$ of entire complex
functions ([35] p.978). All of them are Hausdorff locally convex topological
vector spaces, and the pairs $(\phi,\omega)$, $(h,h^{\prime})$,
$(h^{\prime},h)$, $(\omega,\phi)$ are strong dual to each-other, and Köthe
duals to each-other. The spaces $\phi$, $h$, $\omega$ are metrizable, and thus
Fréchet. The spaces $\phi$, $h$, $\omega$ have the so-called “h” property,
because of which they are nuclear ([35] Chapter2.5). Therefore, $\phi$, $h$,
$\omega$ are NF, and $h^{\prime}$ is DNF. Specially, the space $h$ also has
the countably Hilbert property.

  3. (iii)

It is shown in [35] Chapter3.3 that specially for the sequence spaces
$\lambda=\phi$ or $\lambda=h$ or $\lambda=\omega$, the tensor algebra
$\mathcal{T}(\mathbb{F},\lambda,\mathfrak{p})$ is independent of the choice of
the representant $\mathfrak{p}$ of the topology defining Hilbertian seminorms
on $\mathbb{F}$, thus one may write merely $\mathcal{T}(\mathbb{F},\lambda)$
instead. Moreover, they inherit the NF property of $\mathbb{F}$, making
$\mathcal{T}(\mathbb{F},\lambda)$ a unital associative algebra with jointly
continuous multiplication, with NF topology.

  4. (iv)

It is rather easy to see that $\mathcal{T}(\mathbb{F},\phi)$ is simply the
algebraic tensor algebra $\mathcal{T}_{a}(\mathbb{F})$ with its natural
locally convex direct sum topology, $\mathcal{T}(\mathbb{F},\omega)$ is simply
the Tychonoff tensor algebra $\mathcal{T}(\mathbb{F})$ with its natural
Tychonoff topology.

  5. (v)

For $\lambda$ of the above types, it is shown in [35] Chapter3.3 that the
topology defining seminorms on $\mathcal{T}(\mathbb{F},\lambda)$ may be chosen
to be Hilbertian seminorms

$\displaystyle G$ $\displaystyle\mapsto$
$\displaystyle\sqrt{\sum_{n=0}^{\infty}|u_{n}|^{2}\;(\|G^{(n)}|^{{\otimes}n})^{2}}\qquad({u\in\lambda\\!^{\times},\;\|{\cdot}|\in\mathfrak{p}}),$
(26)

where $\mathfrak{p}$ is a representant of a topology defining family of
Hilbertian seminorms on $\mathbb{F}$. From this, it is explicitely seen that
whenever $\mathbb{F}$ is an NF space admitting a continuous Hilbertian norm,
then the NF space $\mathcal{T}(\mathbb{F},h)$ also admits a continuous
Hilbertian norm.

  6. (vi)

The explicit form of a representant of a topology defining countable family of
increasing Hilbertian seminorms on $\mathcal{T}(\mathbb{F},h)$, encoding its
NF topology, can be given by:

$\displaystyle G$ $\displaystyle\mapsto$
$\displaystyle\sqrt{\sum_{n=0}^{\infty}2^{m\,n}\;(\|G^{(n)}|_{m}^{{\otimes}n})^{2}}\qquad(m\in\mathbb{N}_{0}),$
(27)

where $\|{\cdot}|_{0},\,\|{\cdot}|_{1},\,{\dots}$ is a representant of a
topology defining countable system of increasing Hilbertian seminorms on
$\mathbb{F}$, defining its NF topology. (The formula is the consequence of
[35] Proposition3.7, but is also explicitely used in [36].) From the above
formula it is seen that whenever $\mathbb{F}$ is of CH type, then
$\mathcal{T}(\mathbb{F},h)$ is also of CH type (see also
[22]-RemarkLABEL:S-remCH). We will use the abbreviation
$\mathcal{T}_{h}(\mathbb{F}):=\mathcal{T}(\mathbb{F},h)$, and will call it the
analytic tensor algebra of $\mathbb{F}$, since the topology defining sequence
space $h$ is isomorphic to the space of entire functions $H(\mathbb{C})$.

  7. (vii)

From Eq.(27) it is trivially read off that the counit map
$b:\,\mathcal{T}_{h}(\mathbb{F})\rightarrow\mathbb{R},\,G\mapsto
b\,G:=G^{(0)}$ is continuous. Therefore, the corresponding projection operator
$\mathds{1}\,b$ onto the scalar sector and its complement $I{-}\mathds{1}\,b$
is also continuous. Moreover, the pertinent complementing projection operators
$\mathds{1}\,b$ and $I{-}\mathds{1}\,b$ are orthogonal projections with
respect to the representants of Hilbertian sesquilinear forms from Eq.(27),
and $b\,G=\left<\mathds{1},G\right>_{m}$ holds for all
$G\in\mathcal{T}_{h}(\mathbb{F})$ and all $m\in\mathbb{N}_{0}$.

  8. (viii)

From the Eq.(27) form of the Hilbertian seminorms on
$\mathcal{T}_{h}(\mathbb{F})$ it is seen that this representant family has the
property that whenever $\mathbb{F}$ is CH type NF space, then if its
representant Hilbertian norm family is chosen to be such that the adjacent
norms are nuclear against each-other, the adjacent Hilbertian norms defined by
Eq.(27) are also nuclear against each-other. Similarly, whenever the adjacent
norms on $\mathbb{F}$ are Hilbert–Schmidt against each-other, then the
adjacent norms Eq.(27) are also Hilbert–Schmidt against each-other.

In order to state our existence condition for the solutions of the regularized
MDS equation, we will need to reconsider the space of field correlators to be
based on $\mathcal{T}_{h}(\mathbb{F})$ with the analytic topology, and not on
$\mathcal{T}(\mathbb{F})$ with the Tychonoff direct sum topology. The reason
is that for the construction to work, we need the eventual CH property of
$\mathbb{F}$ to be inherited by its tensor algebra. Therefore, from this point
on, the field algebra $A(\mathbb{F}_{(\mathbb{C})})$ will be defined to be the
appropriately symmetrized subspace of
$\mathcal{T}_{h}(\mathbb{F}_{(\mathbb{C})})$ instead of
$\mathcal{T}(\mathbb{F}_{(\mathbb{C})})$ (see again Remark6 and Definition7
for the technical construction of the symmetrized algebra).

######  Remark 31.

It is worth to verify that the tensor algebra topology on the new field
algebra $A(\mathbb{F}_{(\mathbb{C})})$, inherited from
$\mathcal{T}_{h}(\mathbb{F}_{(\mathbb{C})})$, is not overly strict. For
instance, one would like a typical solution of the regularized MDS equation
for a non-interacting theory to be not excluded from our new, smaller field
algebra $A(\mathbb{F}_{(\mathbb{C})})$. By recalling Remark28, one sees from
Eq.(27) that the pertinent existent solution of the regularized MDS equation
for a non-interacting theory indeed resides in the new, stricter field algebra
as well.

In order to state an existence condition, we shall assume, like in Remark29,
that the base manifold $\mathcal{M}$ under the concrete theory is compact
(with or without boundary, and if with boundary, we assume the cone condition,
so that Sobolev and Maurin theorems hold, see [22]-RemarkLABEL:S-remSobolev).
As stated before, this is a realistic assumption in a conformally invariant
theory, in which case the theory can be re-defined over a compact manifold
with boundary (the conformal compactification of the would-be-spacetime).

######  Remark 32.

Assume that the base manifold $\mathcal{M}$ of the model is compact and its
boundary, if not empty, has the cone condition. Then, the followings hold.

  1. (i)

With such assumption, $\mathbb{F}$ and $\mathbb{F}_{\\!{}_{T}}$ become
countably Hilbert type NF spaces, which is then inherited by
$\mathcal{T}_{h}(\mathbb{F}_{(\mathbb{C})})$, and thus by the field algebra
$A(\mathbb{F}_{(\mathbb{C})})$. From now on, let us use the abbreviation
$\mathcal{H}:=A(\mathbb{F}_{(\mathbb{C})})$.

  2. (ii)

In its original definition, the regularized MDS operator was a separetely
continuous bilinear map
$\mathbf{M}_{\psi_{0}}:\,\mathcal{H}\times\mathbb{F}_{\\!{}_{T}}\rightarrow\mathcal{H},\,(G,\delta\\!\psi_{\\!{}_{T}})\mapsto\mathbf{M}_{\psi_{0},\delta\\!\psi_{\\!{}_{T}}}G$
(see also the original definition Eq.(23), we suppress $\hbar$ and the fixed
mollifying kernel in the notation in this chapter). Due to the compactness
assumption on $\mathcal{M}$, the space $\mathbb{F}_{\\!{}_{T}}$ becomes also
metrizable, therefore the map $\mathbf{M}_{\psi_{0}}$ becomes jointly
continuous ([22]-RemarkLABEL:S-remJoint). Therefore, the regularized MDS
operator, may be also viewed as a continuous linear map
$\mathbf{M}_{\psi_{0}}:\,\mathcal{H}\otimes\mathbb{F}_{\\!{}_{T}}\rightarrow\mathcal{H}$.

  3. (iii)

Due to our compactness assumption on the base manifold $\mathcal{M}$, both
$\mathbb{F}_{\\!{}_{T}}$ and $\mathcal{H}$ became countably Hilbert NF spaces,
which technically means that in both spaces as well as on their tensor
product, the properties Remark29(iii) hold. Denote an associated chain of
Hilbert spaces subordinate to $\mathcal{H}$ by $H_{0},\,H_{1},\,{\dots}$,
their Hilbertian norms by $\|{\cdot}\|_{0},\,\|{\cdot}\|_{1},\,{\dots}$ and
their Hilbertian scalar products by
$\left<\cdot,\cdot\right>_{0},\,\left<\cdot,\cdot\right>_{1},\,{\dots}$.
Similarly, for $\mathbb{F}_{\\!{}_{T}}$ denote by $F_{0},\,F_{1},\,{\dots}$ an
associated chain of Hilbert spaces, their corresponding Hilbertian norms by
$\|{\cdot}\|^{F}_{0},\,\|{\cdot}\|^{F}_{1},\,{\dots}$, and their Hilbertian
scalar products by
$\left<\cdot,\cdot\right>^{F}_{0},\,\left<\cdot,\cdot\right>^{F}_{1},\,{\dots}$.
The associated chain of Hilbert spaces subordinate to
$\mathcal{H}\otimes\mathbb{F}_{\\!{}_{T}}$ can be taken to be the
Hilbert–Schmidt tensor product of the spaces
$H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}$
($m,n\in\mathbb{N}_{0}$), with their canonical Hilbertian cross-norms and
crossed Hilbertian scalar products. (Eventually, a subfamily of this, with
strictly growing norms may also be considered instead.)

  4. (iv)

Because of the nuclearity of the spaces $\mathcal{H}$ and
$\mathbb{F}_{\\!{}_{T}}$, each Hilbertian norm in the above chains will have a
stronger norm in the chain for which the embedding map becomes
Hilbert–Schmidt, and eventually becomes nuclear, for large enough norms in the
chain. (This can also be seen less abstractly on our concrete spaces as a
consequence of the Maurin embedding theorem
[22]-RemarkLABEL:S-remSobolev(LABEL:S-remSobolevII).)

  5. (v)

The continuity of the linear map
$\mathbf{M}_{\psi_{0}}:\,\mathcal{H}\otimes\mathbb{F}_{\\!{}_{T}}\rightarrow\mathcal{H}$
in terms of these Hilbert space chains means that

$\displaystyle\Big{.}\forall k\in\mathbb{N}_{0}:\;\exists
m_{k},n_{k}\in\mathbb{N}_{0}:\;\exists C_{k,m_{k},n_{k}}\in\mathbb{R}^{+}:$
(28) $\displaystyle\Big{.}\forall(G,\delta\\!\psi_{\\!{}_{T}})\in
G\times\mathbb{F}_{\\!{}_{T}}:\quad\|\mathbf{M}_{\psi_{0}}(G\otimes\delta\\!\psi_{\\!{}_{T}})\|_{k}\leq
C_{k,m_{k},n_{k}}\,\|G\|_{m_{k}}\,\|\delta\\!\psi_{\\!{}_{T}}\|_{n_{k}}$ (29)

holds. Since the norms were ordered, the above identity implies that once it
holds, it holds for all $m\geq m_{k}$ and $n\geq n_{k}$ as well with some
constants $C_{k,m,n}\in\mathbb{R}^{+}$. That is, the map
$\mathbf{M}_{\psi_{0}}$ is a continuous linear map
$(\mathcal{H}{\otimes}\mathbb{F}_{\\!{}_{T}})\cap(H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n})\rightarrow\mathcal{H}\cap
H_{k}$ for large enough indices $m,n\in\mathbb{N}_{0}$, given the index
$k\in\mathbb{N}_{0}$. The continuous extension of the map
$\mathbf{M}_{\psi_{0}}$ will be denoted by the same symbol for brevity, and it
is then a continuous linear map
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{k}$, for such indices.

  6. (vi)

By means of (iv), between distant enough indices, the inclusion maps
$H_{m}\supset{\dots}\supset H_{m^{\prime}}$ ($m<m^{\prime}$) and
$F_{n}\supset{\dots}\supset F_{n^{\prime}}$ ($n<n^{\prime}$) become
Hilbert–Schmidt, and eventually become nuclear. Therefore, given
$k\in\mathbb{N}_{0}$, for large enough indices $m,n\in\mathbb{N}_{0}$ the map
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{k}$ becomes Hilbert–Schmidt, and eventually becomes nuclear. (In concrete
spaces, Maurin embedding theorem gives the concrete index bounds, see
[22]-RemarkLABEL:S-remSobolev(LABEL:S-remSobolevII).)

  7. (vii)

As a particular case of the above statement, for all large enough indices
$m,n\in\mathbb{N}_{0}$ one has that the linear map
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{0}$ is Hilbert–Schmidt. The adjoint of this map
$\mathbf{M}_{\psi_{0}}^{\dagger}:\,H_{0}\rightarrow
H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}$ is then also
Hilbert–Schmidt. Therefore, the operator
$\mathbf{M}_{\psi_{0}}^{\dagger}\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}$, becomes a positive
nuclear (trace class) operator.

  8. (viii)

Fix a complete orthonormal basis $(e_{i})_{i\in I}$ in $F_{n}$ (since $F_{n}$
is separable, one may set $I\equiv\mathbb{N}$). Then, for all $G\in H_{m}$ the
estimate

$\displaystyle B(G,G):=\sum_{i\in
I}\left<(G{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}e_{i}),\,\mathbf{M}_{\psi_{0}}^{\dagger}\mathbf{M}_{\psi_{0}}(G{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}e_{i})\right>_{H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}}$
$\displaystyle<$ $\displaystyle\infty$ (30)

is valid. That is because of the Hilbert–Schmidt property of the map
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{0}$. Namely, for some (and therefore: for any) complete orthonormal basis
$(g_{j})_{j\in J}$ in $H_{m}$, one has that $\sum_{j\in J}\sum_{i\in
I}\|\mathbf{M}_{\psi_{0}}(g_{j}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}e_{i})\|_{H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}}^{2}<\infty$
holds (one may set $J\equiv\mathbb{N}$ as well, due to the separability of
$H_{m}$). Taking specially an orthonormal basis $(g_{j})_{j\in J}$ in $H_{m}$,
such that one of its elements is $G/\|G\|_{H_{m}}$, one infers that indeed the
estimate Eq.(30) holds.

  9. (ix)

Given $G\in H_{m}$, the corresponding expression Eq.(30) is independent of the
chosen complete orthonormal basis $(e_{i})_{i\in I}$ in $F_{n}$. That is
because for a Hilbert–Schmidt operator $A$ and an unitary operator $U$ in a
Hilbert space, one has that the Hilbert–Schmidt norm of $A$ and
$U^{\dagger}AU$ is the same.

  10. (x)

Due to the Hilbert–Schmidt property of
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{0}$, the quadratic form $H_{m}\rightarrow\mathbb{C},\,G\mapsto B(G,G)$ is
continuous, and therefore by the polarization formula it gives rise to a
corresponding continuous sesquilinear form $H_{m}\times
H_{m}\rightarrow\mathbb{C},\,(G_{1},G_{2})\mapsto B(G_{1},G_{2})$. Therefore,
by Riesz representation theorem, there is a corresponding unique continuous
linear map $\hat{\mathbf{M}}^{2}_{\psi_{0}}:\,H_{m}\rightarrow H_{m}$, such
that for all $G_{1},G_{2}\in H_{m}$, the identity
$B(G_{1},G_{2})=\big{<}G_{1},\hat{\mathbf{M}}^{2}_{\psi_{0}}G_{2}\big{>}_{H_{m}}$
holds.

  11. (xi)

Due to the positive semidefiniteness of $B$, the map
$\hat{\mathbf{M}}^{2}_{\psi_{0}}$ is a positive operator. Moreover, due to the
Hilbert–Schmidt property of
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{0}$, the map $\hat{\mathbf{M}}^{2}_{\psi_{0}}$ is nuclear (trace class).
The nuclear (trace) norm of
$\hat{\mathbf{M}}^{2}_{\psi_{0}}:\,H_{m}\rightarrow H_{m}$, by construction,
equals to the Hilbert–Schmidt norm of
$\mathbf{M}_{\psi_{0}}:\,H_{m}{\mathbin{{\otimes}\\!_{{}_{\mathrm{HS}}}}}F_{n}\rightarrow
H_{0}$. One can see that the operator $\hat{\mathbf{M}}^{2}_{\psi_{0}}$ is
simply the absolute value squared version of the MDS operator, with its
$\mathbb{F}_{\\!{}_{T}}$ variable traced out.

  12. (xii)

It is obvious from the construction that
$\mathop{\bigcap}\limits_{\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}}\mathop{\mathrm{Ker}}\nolimits(\mathbf{M}_{\psi_{0},\delta\\!\psi_{\\!{}_{T}}})=\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}^{2}_{\psi_{0}})$.

If the reference field $\psi_{0}\in F$ was chosen to be such that it satisfies
the EL equation Eq.(9), then one has that
$\mathbf{M}_{\psi_{0},\delta\\!\psi_{\\!{}_{T}}}\,\mathds{1}=-\mathrm{i}\,L_{\delta\\!\psi_{\\!{}_{T}}}\,\mathds{1}$
($\forall\delta\\!\psi_{\\!{}_{T}}\in\mathbb{F}_{\\!{}_{T}}$). Because of
that, in this situation,
$b\,\hat{\mathbf{M}}_{\psi_{0}}^{2}\,\mathds{1}=\big{<}\mathds{1},\hat{\mathbf{M}}_{\psi_{0}}^{2}\mathds{1}\big{>}_{m}>0$,
and therefore
$\mathds{1}\not\in\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})$.
Thus, generally, the trivial correlator $\mathds{1}$ cannot be a solution of
the regularized MDS equation. One could still aim to find a projection of
$\mathds{1}$ which (up to normalization) satisfies the regularized MDS
equation. Let us denote the orthogonal projection onto
$\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})$ in $H_{m}$
by $P$. Then,
$\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})=\mathop{\mathrm{Ran}}\nolimits(P)$.
One can state the following theorem on $P\mathds{1}$.

######  Theorem 33.

Let $P$ denote the orthoprojection in $H_{m}$ onto
$\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})$. Then, the
following statements are equivalent.

  1. (i)

The solution space of the regularized MDS equation in $H_{m}$ is not empty.

  2. (ii)

One has that $P\mathds{1}\neq 0$.

  3. (iii)

One has that $b\,P\mathds{1}\neq 0$.

######  Proof .

By construction, the MDS equation has solutions in $H_{m}$ if and only if
$b\,\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})\neq\\{0\\}$,
i.e. if and only if there exists some $G\in H_{m}$, such that $b\,P\,G\neq 0$.
(That is because
$\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})=\mathop{\mathrm{Ran}}\nolimits(P)$
and because Remark32(xii).) However, the identity
$b\,P\,G\;=\;\left<\mathds{1},P\,G\right>_{m}\;=\;\left<P\mathds{1},G\right>_{m}$
holds, because of Remark30(vii), and because $P$ was orthoprojection in
$H_{m}$. Therefore, (i) $\Leftrightarrow$ (ii).

Moreover, one has that
$\left<P\mathds{1},P\mathds{1}\right>_{m}\;=\;\left<\mathds{1},P\mathds{1}\right>_{m}\;=\;b\,P\mathds{1}$,
since $P$ was an orthoprojection in $H_{m}$, and because of Remark30(vii).
Therefore, (ii) $\Leftrightarrow$ (iii).

It is seen that the orthoprojection $P$ in $H_{m}$ onto
$\mathop{\mathrm{Ker}}\nolimits(\hat{\mathbf{M}}_{\psi_{0}}^{2})$ plays an
important role in the problematics of existence of MDS solutions. One can
approximate $P$ as below.

######  Theorem 34.

For all $T>0$ parameter, which is not smaller than the operator norm of
$\hat{\mathbf{M}}_{\psi_{0}}^{2}$, and with the notation
$\mathcal{P}:=I-T^{-1}\,\hat{\mathbf{M}}_{\psi_{0}}^{2}$, the operator
sequence $k\mapsto\mathcal{P}^{k}$ converges strongly (pointwise) to $P$ in
$H_{m}$.

######  Proof .

The operator $\mathcal{P}=I-T^{-1}\,\hat{\mathbf{M}}_{\psi_{0}}^{2}$ is a
positive continuous operator with spectrum in $[0,1]$. Therefore,
$k\mapsto\mathcal{P}^{k}$ is a monotonically decreasing sequence of such
operators, bounded from below by the zero operator. Therefore the sequence
$k\mapsto\mathcal{P}^{k}$ converges strongly (pointwise). Since it converges
strongly, it converges also weakly (i.e. matrix element-wise), and its weak
limit equals to the strong limit. We evaluate its strong limit via evaluating
its weak limit, below.

Take any $f,g\in H_{m}$, then there exists a unique complex valued bounded
variation Radon measure $\mu_{\mathcal{P},f,g}$ over $\mathbb{C}$ with
$\mathrm{supp}(\mu_{\mathcal{P},f,g})\subset\mathrm{Sp}(\mathcal{P})\subset[0,1]$,
and
$\left<f,\mathcal{P}g\right>_{m}=\int\limits_{\lambda\in[0,1]}\lambda\,\mathrm{d}\mu_{\mathcal{P},f,g}(\lambda)$
holds. Moreover, for any non-negative integer $k$, one has that
$\left<f,\mathcal{P}^{k}g\right>_{m}=\int\limits_{\lambda\in[0,1]}\lambda^{k}\,\mathrm{d}\mu_{\mathcal{P},f,g}(\lambda)$
holds. One has that
$\int\limits_{\lambda\in[0,1]}\lambda^{k}\,\mathrm{d}\mu_{\mathcal{P},f,g}(\lambda)=\int\limits_{\lambda\in[0,1[}\lambda^{k}\,\mathrm{d}\mu_{\mathcal{P},f,g}(\lambda)+\int\limits_{\lambda\in\\{1\\}}\lambda^{k}\,\mathrm{d}\mu_{\mathcal{P},f,g}(\lambda)$,
where the second term equals to $\left<f,P\,g\right>_{m}$ by construction. The
function $\lambda\mapsto\lambda^{k}$ converges to zero pointwise on $[0,1[$,
and is bounded by the constant $1$ function which is $\mu_{\mathcal{P},f,g}$
absolute integrable on $[0,1[$. Therefore, by Lebesgue’s theorem of dominated
convergence, the integral
$\int\limits_{\lambda\in[0,1[}\lambda^{k}\,\mathrm{d}\mu_{\mathcal{P},f,g}(\lambda)$
tends to zero as a function of $k$. Therefore,
$\left<f,\mathcal{P}^{k}g\right>_{m}$ converges to $\left<f,P\,g\right>_{m}$
in $k$, i.e. $\mathcal{P}^{k}$ converges weakly to $P$.

By combining Theorem33 and Theorem34, one can draw the following conclusion.

######  Corollary 35.

For all $T>0$ parameter, which is not smaller than the operator norm of
$\hat{\mathbf{M}}_{\psi_{0}}^{2}$, one has that the iteration

$\displaystyle G_{0}:=\mathds{1},\qquad
G_{k+1}:=G_{k}-T^{-1}\,\hat{\mathbf{M}}_{\psi_{0}}^{2}\,G_{k}$ (31)

converges in $H_{m}$. Therefore, there exists the finite real number
$\nu:=\lim\limits_{k\rightarrow\infty}b\,G_{k}\in\mathbb{R}$.
The MDS equation has solutions in $H_{m}$ if and only if $\nu\neq 0$.
Moreover, if $\nu\neq 0$, then
$\frac{1}{\nu}\,\lim\limits_{k\rightarrow\infty}G_{k}$ is a solution of the
MDS equation in $H_{m}$.

######  Proof .

Clearly, by construction we have that $G_{k}=\mathcal{P}^{k}\,\mathds{1}$ for
all $k\in\mathbb{N}_{0}$, and we have just shown in Theorem34, that
$k\mapsto\mathcal{P}^{k}$ converges strongly to $P$. Therefore,
$\lim\limits_{k\rightarrow\infty}G_{k}=P\,\mathds{1}$, and
$\nu=b\,P\,\mathds{1}$. Applying then Theorem33, we get the stated result.

######  Remark 36.

The following identities are useful for technical evaluation.

  1. (i)

The minimal factor $T>0$, which can be used in the above existence test is the
operator norm of $\hat{\mathbf{M}}^{2}_{\psi_{0}}$. The operator norms are
generally hard to estimate. However, it can be estimated from above by the
trace norm of $\hat{\mathbf{M}}^{2}_{\psi_{0}}$, or equivalently, by the
Hilbert–Schmidt norm of $\mathbf{M}_{\psi_{0}}$, which are technically easier
to evaluate.

  2. (ii)

It is a useful fact that the indicator sequence
$k\mapsto\nu_{k}:=b\,G_{k}\in\mathbb{R}$ consists of non-negative numbers, and
is monotonically decreasing. That is because
$\nu_{k}=b\,G_{k}=\left<\mathds{1},G_{k}\right>_{m}=\left<\mathds{1},\mathcal{P}^{k}\mathds{1}\right>_{m}$.
The operator sequence $k\mapsto\mathcal{P}^{k}$ consists of a sequence of
positive operators, which are monotonically decreasing. Therefore
$k\mapsto\nu_{k}$ inherits this property. Thus, it is enough to test whether
the indicator sequence $k\mapsto\nu_{k}$ is bounded away from zero. Moreover,
the scalar component $b\,G_{k}$ of the approximants $G_{k}$ start from $1$,
they do stay real, and they do not flip sign from positive to negative, and
they monotonically decrease.

  3. (iii)

By means of Corollary35, for concrete models, evaluating whether the indicator
$\nu$ is bounded away from zero, is expected to involve elaborate Sobolev
estimates. The corollary, however, pinpoints a well defined point where one
has to invoke these estimates, and therefore this can be considered as a
useful existence test condition.

  4. (iv)

Since Corollary35 is a necessary and sufficient condition, and not merely a
sufficient condition, one may also use it in the reverse direction. Namely, if
for a concrete model the regularized MDS equation had any solutions, then the
iteration scheme of Corollary35 is guaranteed to be good enough to be
convergent, and to produce one particular MDS solution. This is a useful piece
of information, even without actually performing the above Sobolev estimates.

##  6 Concluding remarks

In the QFT literature, the master Dyson–Schwinger (MDS) equation on the field
correlators is known to be a differential reformulation of the Feynman
integral formalism. In this paper it is shown that the MDS equation can be
cast into a particular presentation, in which the involved function spaces and
operators are perfectly well defined, regardless of a fixed background
spacetime metric, or causal structure, or signature. Moreover, the Wilsonian
regularized version of the construction is also shown to be well defined in
such a generally covariant setting. A necessary and sufficient condition is
proved for the solution space of the regularized MDS equation to be nonempty,
for conformally invariant Lagrangians. The pertinent theorem is constructive
in the sense that it provides an iterative algorithm to obtain an MDS
solution. The algorithm is guaranteed to converge whenever the solution space
is nonempty, and could be eventually used for a lattice QFT-like
nonperturbative numerical solution scheme, capable of working in the original
metric signature.

## Acknowledgments

We would like to thank to the organizers of the Simplicity III workshop at the
Perimeter Institute, and especially to Neil Turok and Job Feldbrugge for the
inspiring exposition on the problematics of Feynman integral formulation in
Lorentz signature, which inspired this work. We would also like to thank to
Antal Jakovác for enlightening discussion on Feyman integral formulation and
ERGE from the physical point of view. We would also like to thank Áron Szabó,
Bence Racskó and Igor Khavkine for valuable feedback on the mathematical
content of the manuscript. Special thanks to János Kristóf for the
enlightening mathematical inputs concerning the theory of topological vector
spaces (TVS) and measure theory on them, moreover to Zsigmond Tarcsay for
double-checking the mathematical content of the paper concerning the theory of
TVS, especially regarding the questions of operator closability.

This work was supported in part by the Hungarian Scientific Research fund
(NKFIH K-138152).

##  Appendix A Continuity properties of the Euler–Lagrange functional

Our presentation of the master Dyson–Schwinger (MDS) operator heavily relies
on the precise definition of the Euler–Lagrange functional of a classical
field theory. In order to pin down the topological properties of the involved
spaces and precise continuity property of their operators, we need to briefly
recall the standard variational formulation of a classical field theory. For
the sake of simplified treatment, we will use neither the jet formalism, nor
the theory of general connections over fiber bundles [37, 38]. We will rather
concentrate on the TVS theory side [22], i.e. we keep the differential
geometric treatment to a reasonable appropriate minimum. Let $\mathcal{M}$
denote throughout the paper a finite dimensional real smooth orientable and
oriented manifold, and let $m:=\dim(\mathcal{M})$. It may be compact or
noncompact, and may be with or without boundary (if with boundary, we assume
the cone condition for it, so that locally the Sobolev and Maurin embedding
theorems hold, see also [22]-SectionLABEL:S-subsecSobolev). The manifold
$\mathcal{M}$ is meant to model the spacetime manifold, or eventually, the
compact manifold with boundary underlying the conformal compactification
(Penrose diagram) of a spacetime. Let $V(\mathcal{M})$ be some real vector
bundle over $\mathcal{M}$ with finite dimensional fibers. Denote, as usual, by
$\Gamma\big{(}\cdot\big{)}$ the space of smooth sections. In particular, the
space $\Gamma\big{(}V(\mathcal{M})\big{)}$ denotes the real vector space of
smooth sections of $V(\mathcal{M})$ (these are meant to model the matter
fields). The covariant derivation operators over
$\Gamma\big{(}V(\mathcal{M})\big{)}$ (which are meant to model the mediator
fields) form an affine space with subordinate vector space
$\Gamma\big{(}T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})\big{)}$,
as it is common knowledge. More particularly, covariant derivation operators
can be considered as sections of an affine bundle over $\mathcal{M}$, which we
will denote by $D\\!V(\mathcal{M})$, and then a covariant derivation is an
element of $\Gamma\big{(}D\\!V(\mathcal{M})\big{)}$. The bundle
$D\\!V(\mathcal{M})$ is an affine bundle with subordinate vector bundle
$T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})$, the
sections of which are the difference tensors (or gauge potentials, in field
theory). As usual, the symbols $\otimes$ shall denote pointwise tensor product
of vector bundles over the same base, whereas $\oplus$ will denote pointwise
direct sum of vector bundles over the same base. The fibered product
(pointwise cartesian product, or Whitney sum) of two generic fiber bundle over
the same base shall be denoted by
$V(\mathcal{M})\mathbin{{\times}_{\\!\\!{}_{W}}}U(\mathcal{M})$, and their
elements by $(v,u)_{\\!\\!{}_{W}}$ in order to indicate that these are pairs
of fields, over the same base points. The subscript ${}_{{}_{W}}$ is used in
order to distinguish the above from sections $(v,u)$ of the cartesian product
bundle $V(\mathcal{M})\times U(\mathcal{M}^{\prime})$, which would be a bundle
over the product manifold $\mathcal{M}{\times}\mathcal{M}^{\prime}$
(eventually $\mathcal{M}^{\prime}=\mathcal{M}$).

######  Definition 37.

Let $\mathcal{M}$, $V(\mathcal{M})$ and $D\\!V(\mathcal{M})$ as above.

A Lagrange form is a base point preserving, smooth fiber bundle homomorphism

$\displaystyle\mathrm{L}:\;V(\mathcal{M})\;\oplus\;T^{*}(\mathcal{M}){\otimes}V(\mathcal{M})\;\oplus\;T^{*}(\mathcal{M}){\wedge}T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})\;\longrightarrow\;\mathop{\wedge}\limits^{m}T^{*}(\mathcal{M}).$

By construction, a Lagrange form takes some sections

$\displaystyle v\in\Gamma\big{(}V(\mathcal{M})\big{)},\quad
D\\!v\in\Gamma\big{(}T^{*}(\mathcal{M}){\otimes}V(\mathcal{M})\big{)},\quad
P\in\Gamma\big{(}T^{*}(\mathcal{M}){\wedge}T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})\big{)}$

into a maximal form field
$\mathrm{L}(v,D\\!v,P)\in\Gamma\big{(}\mathop{\wedge}\limits^{m}T^{*}(\mathcal{M})\big{)}$.

An element
$(v,\nabla)_{\\!\\!{}_{W}}\in\Gamma\big{(}V(\mathcal{M})\mathbin{{\times}_{\\!\\!{}_{W}}}D\\!V(\mathcal{M})\big{)}$
is called a field configuration. The field configurations form an affine space
over the real vector space ${\Gamma\big{(}V(\mathcal{M})\oplus
T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})\big{)}}$.
An element $(\delta\\!v,\delta\\!C)_{\\!\\!{}_{W}}$ from that space is called
a field variation.

The map

$\displaystyle\Gamma\big{(}V(\mathcal{M})\mathbin{{\times}_{\\!\\!{}_{W}}}D\\!V(\mathcal{M})\big{)}\;\longrightarrow\;\Gamma\big{(}\mathop{\wedge}\limits^{m}T^{*}(\mathcal{M})\big{)},\quad(v,\nabla)_{\\!\\!{}_{W}}\;\longmapsto\;\mathrm{L}(v,\nabla{v},P(\nabla))$

is called the Lagrangian expression, where $\nabla{v}$ is the covariant
derivative of the section $v$, and $P(\nabla)$ is the curvature tensor of
$\nabla$. (Note that the expression $(v,\nabla{v},P(\nabla))_{\\!\\!{}_{W}}$
ecodes the same information as the first jet of a field configuration
$(v,\nabla)_{\\!\\!{}_{W}}$, but we do not intend to use the jet formalism in
the present paper.)

Given a Lagrange form $\mathrm{L}$, its action functional is the real Radon
measure valued map

$\displaystyle
S^{\mathrm{L}}:\quad\Gamma\big{(}V(\mathcal{M})\mathbin{{\times}_{\\!\\!{}_{W}}}D\\!V(\mathcal{M})\big{)}\;\longrightarrow\;\mathrm{Rad}(\mathcal{M},\mathbb{R}),\quad(v,\nabla)_{\\!\\!{}_{W}}\;\longmapsto\;S^{\mathrm{L}}(v,\nabla)_{\\!\\!{}_{W}}$

where on compact subsets $\mathcal{K}\subset\mathcal{M}$ the definition is
$S^{\mathrm{L}}_{\mathcal{K}}(v,\nabla)_{\\!\\!{}_{W}}:=\int\limits_{\mathcal{K}}\mathrm{L}(v,\nabla{v},P(\nabla))$,
i.e. the action functional is the Radon measure defined by local integrals of
the Lagrangian expression, as usual.

We use the shorthand notation
$F:=\Gamma\big{(}V(\mathcal{M})\mathbin{{\times}_{\\!\\!{}_{W}}}D\\!V(\mathcal{M})\big{)}$
for the space of field configurations, moreover
$\mathbb{F}:=\Gamma\big{(}V(\mathcal{M})\,\oplus\,T^{*}(\mathcal{M}){\otimes}V(\mathcal{M}){\otimes}V^{*}(\mathcal{M})\big{)}$
for the space of field variations. The space $F$ is an affine space over the
real vector space $\mathbb{F}$. The real vector space $\mathbb{F}$ may be
naturally endowed with the standard $\mathcal{E}$ smooth function topology.
(The $\mathcal{E}$ topology is defined by the family of arbitrary order
Sobolev norms of over compact patches of $\mathcal{M}$.) With this topology
$\mathbb{F}$ and thus $F$ become Hausdorff locally convex topological vector
and affine spaces, respectively. It is also common knowledge
[22]-RemarkLABEL:S-remNuclear), that $\mathbb{F}$ with the $\mathcal{E}$
topology becomes a _nuclear Fréchet space_ , which fact will be an important
detail in the QFT construction.

The real vector space of real valued Radon measures
$\mathrm{Rad}(\mathcal{M},\mathbb{R})$ can be also naturally endowed with a
topology, defined by compact setwise total variations as family of seminorms,
or equivalently, by the convergence of measure sequences over compact sets.
With this, $\mathrm{Rad}(\mathcal{M},\mathbb{R})$ becomes a Hausdorff locally
convex topological vector space.

######  Remark 38.

It is not true in general that the continuity of a map between topological
spaces is equivalent to its sequential continuity. It is common knowledge,
however, that metrizable topological spaces are _sequential_
([22]-RemarkLABEL:S-remSequential), i.e. their topology is completely
characterized by the convergence of sequences. Since $\mathbb{F}$ is Fréchet
space, by construction its topology is metrizable in a translationally
invariant way, and therefore also is the topology of $F$. In particular, a map
$S:\,F\rightarrow Y$ to any topological space $Y$ is continuous if and only if
$S$ is sequentially continuous, i.e. it maps convergent sequences in $F$ to
convergent sequences in $Y$.

######  Remark 39.

The following can be observed.

  1. (i)

The action functional was defined to be a Radon measure valued map. That was
motivated by the fact that no asymptotics was prescribed on the field
configurations $F$, nor it was assumed that $\mathcal{M}$ is compact. Because
of that, one cannot guarantee that the smooth maximal form field
$\mathrm{L}(v,\nabla{v},P(\nabla))$ is integrable throughout the full
$\mathcal{M}$ for sufficiently many field configurations
$(v,\nabla)_{\\!\\!{}_{W}}\in F$. It is, however, always locally integrable,
hence the action functional as a Radon measure valued map is meaningful, and
everywhere defined.

  2. (ii)

Due to Lebesgue’s theorem of dominated convergence, the action functional is
sequentially continuous, and therefore by means of Remark38, it is continuous.

The action functional is everywhere differentiable in the Fréchet–Hadamard
sense (see also [22]-SectionLABEL:S-subsecFH), as it is common knowledge in
Lagrangian field theory. In order to show its explicit form, we recall some
differential geometric identities. We will use Penrose abstract indices for
the tangent tensors throughout the section.

######  Remark 40.

If $\nabla$ is a covariant derivation over $T(\mathcal{M})$, then there is a
unique covariant derivation $\tilde{\nabla}$ over $T(\mathcal{M})$ associated
to it, having vanishing torsion tensor and having the same affine parametrized
geodesics as $\nabla$. The covariant derivation $\tilde{\nabla}$ is called the
_torsion-free part_ of $\nabla$. In explicit formulae: whenever $v^{b}$ is a
smooth section of $T(\mathcal{M})$, then one has
$\tilde{\nabla}_{a}v^{b}=\nabla_{a}v^{b}+\frac{1}{2}T(\nabla)_{ac}^{b}v^{c}$,
where $T(\nabla)_{ac}^{b}$ denotes the torsion tensor of $\nabla$.

######  Theorem 41.

The action functional $S^{\mathrm{L}}$ is everywhere differentiable, and its
derivative at some fixed $(v,\nabla)_{\\!\\!{}_{W}}\in F$ is a continuous
linear map
$DS^{\mathrm{L}}(v,\nabla)_{\\!\\!{}_{W}}:\,\mathbb{F}\rightarrow\mathrm{Rad}(\mathcal{M},\mathbb{R})$,
given by the formula

$\displaystyle\Big{.}(\delta\\!v,\delta\\!C)_{\\!\\!{}_{W}}\;\mapsto\;\big{(}DS^{\mathrm{L}}_{\mathcal{K}}(v,\nabla)_{\\!\\!{}_{W}}\,\big{|}\,(\delta\\!v,\delta\\!C)_{\\!\\!{}_{W}}\big{)}=$
(32)
$\displaystyle\int\limits_{\mathcal{K}}\Bigl{(}D_{1}\mathrm{L}(v,\nabla{v},P(\nabla))\,\delta\\!v+D_{2}^{a}\mathrm{L}(v,\nabla{v},P(\nabla))\,(\nabla_{a}{\delta\\!v}+\delta\\!C_{a}v)+2\,D_{3}^{[ab]}\mathrm{L}(v,\nabla
v,P(\nabla))\,\tilde{\nabla}_{[a}\delta\\!C_{b]}\Bigr{)},$ (33) (34)

when evaluated on some compact subset $\mathcal{K}\subset\mathcal{M}$. Here,
$D_{1}\mathrm{L}$, $D_{2}\mathrm{L}$, $D_{3}\mathrm{L}$ denote the spacetime
pointwise partial derivative of $\mathrm{L}$ against its first, second and