ileeciRGB_v2.icc

# Møller maps for Dirac fields in external backgrounds

Valentino Abram  Dipartimento di Matematica, Università di Trento, 38123 Povo
(TN), Italy<EMAIL_ADDRESS>and  Romeo Brunetti  Dipartimento di
Matematica, Università di Trento, 38123 Povo (TN), Italy
<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the foundations of the algebraic treatment of classical
and quantum field theories for Dirac fermions under external backgrounds
following the initial contributions made by several collegues. The treatment
is restricted to contractible spacetimes of globally hyperbolic nature in
dimensions $d\geq 4$. In particular, we construct the classical Møller maps
intertwining the configuration spaces of _charged_ and _uncharged_ fermions.
In the last part, as a first step towards a quantization of the theory, we
explore the combination of the classical Møller maps with Hadamard
bidistributions and prove that they are involutive isomorphisms (algebraically
and topologically) between suitable (formal) algebras of functionals
(observables) over the configuration spaces of charged and uncharged Dirac
fields.

###### Contents

  1. 1 Introduction
  2. 2 Geometric and analytic preliminaries
    1. 2.1 Basic notions in spin geometry on pseudo-riemannian manifolds
    2. 2.2 The geometric description of charged spinors
  3. 3 The classical Møller map on field configurations
    1. 3.1 The entwining maps between sections
    2. 3.2 The explicit construction of the classical Møller map
  4. 4 The behaviour of the classical Møller map in the case of a $U(1)$ gauge charge
    1. 4.1 The classical Møller map and Green operators
    2. 4.2 The classical Møller map and Hadamard bidistributions
  5. 5 The classical Møller map on the observable algebras
    1. 5.1 The Poisson $\ast$-algebras of observables
    2. 5.2 Deformation quantization and the classical Møller maps
  6. 6 Conclusions and outlook

##  1\. Introduction

The main aim of this paper is to contribute to a by now well established
framework for the algebraic treatment of quantum systems made of fermions in
arbitrary backgrounds (metrics, external fields etc.) which aims at the
rigorous determination of physical effects (see, _e.g._ , [13, 26, 18, 9, 36,
10, 35, 42, 37, 20, 41, 31, 8]). It is a known fact that especially for strong
space-time dependent external fields the mostly used theoretical frameworks
suffer from several deficits. In general, the lack of space-time symmetries
implies a missing privileged state (vacuum) with the related impossibility to
use familiar tools as Fourier transformations and Fock spaces. All these limit
terribly the ability of physicists to deduce observable effects in these
quantum situations without making further drastic and simplifying assumptions.
Indeed, there exists a very large literature on the subject which is full of
interesting ideas, techniques and results (see, _e.g._ , [16, 17]). Tipically,
however, the proposals are made without referring to general and deep basic
concepts and hence with a lot of _ad hoc_ assumptions. A possible way out of
these difficulties is to refer to recent structural advances in quantum field
theories using the algebraic approach. The new perspective refers to deep
conceptual advancements, for instance local covariance [7], and technically,
instead of Fourier transformations, uses its modern improvement named micro-
local analysis [27], in particular wave front sets [38, 4]. There have been
several recent interesting results that point towards the validity of this
claim. For instance, Fröb and Zahn [20] have shown how to rigorously derive
the trace anomaly for chiral fermions using in particular the rigorous method
of Hadamard subtraction. At variance w.r.t. the literature in physics, this
was done in Lorentzian spacetime, and invoking physical principles as
invariance of the stress-energy tensor to show the cancellation of unwanted
terms on which physicists debated for long.

Our main concern is to build up at first the classical tools that can be used
later to develop the formalism towards the quantum aspects. In the present
paper we concentrate basically only in the former aspects but make an initial
step into the latter. We demonstrate that the classical Møller maps are
involutive isomorphisms for the various classes of mathematical objects of
pertinence for us. Indeed, at first we introduce various spinorial
configurations spaces as sections of diverse bundles over semi-Riemannian
spaces, and prove that the Møller maps are isomorphisms between the charged
and uncharged spinor bundles. Thence, we extend the structure to (nonlinear)
_functionals_ over such bundles forming (involutive) algebras. One should look
at them as the (abelian) algebras of observables of the theory. A first step
forward is done here by the construction of a _Poisson_ algebra. This entails
at first the selection of a “good” subset of functionals. The term good refers
to the fact that in order to rigorously define a Poisson structure for such
field theories one can make a covariant choice which is determined by the use
of the Peierls’ backets (see, _e.g._ , [6]). This implies the use of causal
Green operators (propagators) whose kernels, seen as distributions, do not
directly allow multiplications by generic functionals. It is here that
microlocal analysis appears as the right tool. The good selection is indeed
made out of the desire to define the product of functionals, and their
derivatives, with the propagators. A sufficient criterion for those products
to exist is the Hörmander’s one based on wave front sets. Hence, the so called
_microcausal functionals_ make their crucial appearance here. These are
functionals with prescribed singularities whose wave front sets combine well
with the wave front set of the propagators as to satisfy Hörmander’s
criterion.

We then develop the formalism doing a first step into the quantum realm by
extending the algebras to the formal algebras of deformation quantization, by
changing the classical product with the use of the Hadamard prescription.

In the course of the paper we develop geometric and analytic descriptions by
adopting a practical and precise pointwise view which has the merit of being
rather explicit. The resulting formalism had high advantages which we do hope
compensate the heaviness of notations.

##  2\. Geometric and analytic preliminaries

###  2.1. Basic notions in spin geometry on pseudo-riemannian manifolds

We begin the exposition by recalling some definitions and results in order to
fix the notation used throughout the paper. We refer the reader to [39], [30]
and [33] for more details.

We shall work on $n$-dimensional spacetimes, that is, couples $(M,g)$
consisting of a connected, paracompact, orientable, time-orientable Hausdorff
smooth manifold $M$ and a non-degenerate, pseudo-riemannian metric $g$. We
shall further suppose that two additional conditions hold:

  1. $(i)$

we assume $(M,g)$ is _globally hyperbolic_ : this entails that given a
normally hyperbolic operator, this admits retarded and advanced Green
operators (see, _e.g._ , [3, 1]);

  2. $(ii)$

we also assume that $\text{dim}(M)\geq 4$: this entails that there exists a
universal covering homomorphism
$\xi_{0}\colon\mathrm{Spin}^{0}_{r,1}\to\text{SO}_{r,1}^{0}$ between the
identity component of the $\mathrm{Spin}$ group $\mathrm{Spin}_{r,1}$ and the
identity component of the signature $(r,1)$ of the special orthogonal group
[33, Proposition 12.1.41].

On the manifold $M$ modelling our spacetime, we shall consider _fiber bundles_
, i.e., quadruples $(B,M,\pi,F)$ where $\pi\colon B\to M$ is a smooth,
surjective map and such that there exists an open cover
$\\{U_{\alpha}\\}_{\alpha\in A}$ of the base manifold $M$ and an associated
collection $\\{\phi_{\alpha}\colon\pi^{-1}(U_{\alpha})\to U_{\alpha}\times
F\\}_{\alpha\in A}$ of diffeomorphisms, called _trivialization of the bundle_
, such that

$\pi_{1}\circ\phi_{\alpha}=\pi|_{\pi^{-1}(U_{\alpha})}\ \text{for every}\
\alpha\in A\ .$ (2.1)

Notice that given a point $p\in M$, the fiber
$B_{p}\overset{\cdot}{=}\pi^{-1}(p)$ is diffeomorphic to $F$; this is the
reason $F$ is called _typical fiber_. We can consider the _transition
functions_ of the bundle, that is, the maps

$\begin{split}\phi_{\alpha\beta}\colon(U_{\alpha}\cap U_{\beta})\times
F&\to(U_{\alpha}\cap U_{\beta})\times F\\\
(p,f)\qquad&\mapsto\phi_{\alpha}\left(\phi_{\beta}^{-1}(p,f)\right)\
.\end{split}$

By the condition (2.1), we have that
$\phi_{\alpha\beta}(p,f)=(p,g_{\alpha\beta}(p)(f))$ for some
$g_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\to\text{Aut}(F)$; the
collection $\\{g_{\alpha\beta}\\}_{\alpha,\beta\in A}$ is called _cocycle_.

In the context of quantum field theory, one usually deals with two kinds of
fiber bundles:

  1. $(i)$

_vector bundles_ , that is, fiber bundles whose typical fiber is a vector
space and whose cocycle is such that $g_{\alpha\beta}(p)\in\text{GL}(F)$ for
every $\alpha,\beta\in A$, $p\in U_{\alpha}\cap U_{\beta}$;

  2. $(ii)$

_principal $G$-bundles_, that is, fiber bundles whose typical fiber is the
group $G$, whose total space $P_{G}$ is a right $G$-manifold with right action
$r_{g}\colon P_{G}\to P_{G}$ and whose trivialization
$\\{\phi_{\alpha}\colon\pi^{-1}(U_{\alpha})\to U_{\alpha}\times G\\}$ consists
of right $G$-equivariant maps. This entails that the cocycle
$\\{g_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\to\text{Aut}(G)\\}$
consists of left-translations [39, Lemma 27.7].

Given a principal $G$-bundle $\pi\colon P_{G}\to M$ and a representation
$\rho\colon G\to\text{GL}(V)$ of the group on a finite-dimensional vector
space $V$, we can construct the _associated vector bundle_ $\pi\colon
P_{G}\times_{\rho}V\to M$ with typical fiber $V$. [39, Theorem 31.9] shows
that there exist linear isomorphisms

$\begin{split}&{\cdot}^{\flat}\colon\Omega_{\rho}^{q}(P_{G},V)\to\Gamma\left(\wedge^{q}T^{\ast}M\otimes
P_{G}\times_{\rho}V\right)\qquad{\cdot}^{\sharp}\colon\Gamma\left(\wedge^{q}T^{\ast}M\otimes
P_{G}\times_{\rho}V\right)\to\Omega_{\rho}^{q}(P_{G},V)\end{split}$ (2.2)

between the spaces $\Omega_{\rho}^{q}(P_{G},V)$ of $V$-valued, tensorial
$q$-forms of type $\rho$ on $P_{G}$ and the space of $q$-forms on $M$ with
values in $P_{G}\times_{\rho}V$.

It is a well-known fact that gauge fields are represented by Ehresmann
connections $\omega\in\Omega^{1}(P,\mathfrak{g})$ on a suitable principal
$G$-bundle [32]. These connection 1-forms can be used to induce an _exterior
covariant differentiation_
$D\colon\Omega_{\rho}^{q}(P_{G},V)\to\Omega_{\rho}^{q+1}(P_{G},V)$,

$D\varphi\overset{\cdot}{=}d\varphi+\omega\cdot_{\rho}\varphi$ (2.3)

where

$(\omega\cdot_{\rho}\varphi)_{p}(v_{1},\dots,v_{q+1})=\frac{1}{q!}\sum_{\sigma\in
S_{q+1}}\text{sgn}(\sigma)\rho_{\ast}(\omega_{p}(v_{\sigma(1)}))\varphi_{p}(v_{\sigma(2)},\dots,v_{\sigma(q+1)})\
.$

We can then use the exterior covariant derivative to endow the associated
vector bundle $P_{G}\times_{\rho}V$ with a connection
$\nabla\colon\mathfrak{X}(M)\times\Gamma(P_{G}\times_{\rho}V)\to\Gamma(P_{G}\times_{\rho}V)$,

$\nabla_{v}s\overset{\cdot}{=}(D(s^{\sharp}))^{\flat}(v)\ .$

This section of the associated vector bundle can be expressed locally as

$(\nabla_{v}s)(p)=\left[\sigma(p),v(p)\left(s^{\sharp}\circ\sigma\right)+\rho_{\ast}\Big{(}\omega_{\sigma(p)}\big{(}{\sigma_{\ast}}_{p}(v_{p})\big{)}\Big{)}s^{\sharp}(\sigma(p))\right]$
(2.4)

where $\sigma\colon U\to{P_{G}}_{U}$ is a _local_ section of the principal
$G$-bundle $P_{G}$, usually deemed _local gauge choice_. The pullback
$\sigma^{\ast}\omega\in\Omega^{1}(U,\mathfrak{g})$ is called _local gauge
potential_.

An important construction which can be carried out in the case of complex
vector bundles is that of the _conjugate bundle_. Let us recall that given a
complex vector space $V$, the conjugate vector space $\overline{V}$ is a
vector space which is real-isomorphic to $V$ and whose complex structure is
the conjugate complex structure. The antilinear isomorphism bewteen $V$ and
$\overline{V}$ shall be denoted with $C\colon\overline{V}\to V$, and it an be
used to induce, starting from a linear map $L\colon V\to V$, a linear map
$\overline{L}\colon\overline{V}\to\overline{V}$ by defining

$\overline{L}\overset{\cdot}{=}C^{-1}\circ L\circ C\ .$

Given a basis $\\{\overline{e}_{i}\\}_{i\in I}$ of $\overline{V}$, the
components $\\{\overline{L}^{i}_{j}\\}_{i,j\in I}$ of $\overline{L}$ are the
complex conjugates of the components of $L$ in the basis
$\\{C\overline{e}_{i}\\}_{i\in I}$, i.e.

$\overline{L}^{i}_{j}=\overline{L^{i}_{j}}$

If $V$ is endowed with a sesquilinear form, then one can show that
$\overline{V}\simeq V^{\ast}$. These facts can be extended to the case of
complex vector bundles in the following way: first of all, consider a vector
bundle $E\overset{\pi}{\to}M$ with typical fiber $V$, whose cocycle is given
by $\\{g_{\alpha\beta}\\}_{\alpha,\beta\in A}$; then one can construct another
vector bundle with typical fiber $\overline{V}$ and whose cocycle is given by
$\\{\overline{g_{\alpha\beta}}\\}_{\alpha,\beta\in A}$; this vector bundle is
the conjugate vector bundle $\overline{E}\overset{\pi}{\to}M$. The conjugation
map $C\colon\overline{V}\to V$ can be extended to a vector bundle anti-
isomorphism $C\colon\overline{E}\to E$, and if $E$ is endowed with a hermitian
metric $h$, then

  1. (i)

$h$, which assigns to every point $p\in M$ a sesquilinear form $h_{p}$ on
$E_{p}$, can be understood as a map assigning to each $p\in M$ a bilinear map
$h_{p}\colon E_{p}\times\overline{E}_{p}\to\mathbb{C}$;

  2. (ii)

we have an isomorphism $\overline{E}\simeq E^{\ast}$.

Let us now consider the tangent bundle $\pi\colon TM\to M$ on our spacetime;
by the assumptions on $(M,g)$ this is an oriented vector bundle endowed with a
Lorentzian structure, and we can thus consider its oriented, time-oriented,
pseudo-orthonormal frame bundle, that is, a principal
$\text{SO}_{r,1}^{0}$-bundle $\pi_{P}\colon P_{\text{SO}_{r,1}^{0}}(M)\to M$
naturally associated to it. If we further assume that the second Stiefel-
Whitney class $w_{2}(TM)$ vanishes, we can consider a _spin structure_ on
$TM$, that is, a principal $\mathrm{Spin}_{r,1}^{0}$-bundle
$\pi_{P^{\prime}}\colon P_{\mathrm{Spin}_{r,1}^{0}}(M)\to M$ coupled with a
2-sheeted covering

$\xi\colon P_{\mathrm{Spin}^{0}_{r,1}}\to P_{\mathrm{SO}_{r,1}^{0}}$ (2.5)

such that $\xi\left(r_{g}(p)\right)=r_{\xi_{0}(g)}\left(\xi(p)\right)$ for
every $p\in P_{\mathrm{Spin}_{r,1}^{0}}$, $g\in\mathrm{Spin}_{r,1}^{0}$, with
$\xi_{0}\colon\mathrm{Spin}_{r,1}^{0}\to\text{SO}_{r,1}^{0}$ being the
universal covering map. This entails in particular that the cocycle
$\\{g_{\alpha\beta}\colon U_{\alpha}\cap
U_{\beta}\to\text{Aut}(\text{SO}_{r,1}^{0})\\}$ of the principal
$\text{SO}_{r,1}^{0}$-bundle is given by $\\{\xi_{0}(s_{\alpha\beta})\\}$,
where $\\{s_{\alpha\beta}\colon U_{\alpha}\cap
U_{\beta}\to\text{Aut}(\mathrm{Spin}_{r,1}^{0})\\}$ is a cocycle for the
principal $\mathrm{Spin}_{r,1}^{0}$-bundle.
By the fundamental theorem of Riemannian geometry, there exists a unique
torsion-free, metric-compatible connection
$\omega\in\Omega^{1}(P_{\text{SO}_{r,1}^{0}},\mathfrak{so}_{r,1}^{0})$, the
Levi-Civita connection; using the covering map (2.5), this connection can be
pulled back to a connection 1-form
$\omega^{s}\in\Omega^{1}(P_{\mathrm{Spin}_{r,1}^{0}},\mathfrak{spin}_{r,1}^{0})$,
usually called _spin connection_.

We can then consider the following: depending on the dimension $r+1$ of our
spacetime $M$, there exists a complex representation of the Spin group
$\Delta_{r,1}^{\mathbb{C}}\colon\mathrm{Spin}_{r,1}^{0}\to\text{GL}(V,\mathbb{C})$
which is irreducible in the odd dimensional case and reducible in the even
dimensional one [30, §I, Proposition 5.15]. In both cases, the representation
descends from a complex representation
$\rho\colon\mathscr{C}\ell_{r,1}\to\text{End}(V,\mathbb{C})$ of the Clifford
algebra $\mathscr{C}\ell_{r,1}$. [30, §I, Theorem 5.7] shows that there exists
a unique Clifford algebra representation if $r+1$ is even, while there exist
two inequivalent representations if $r+1$ is odd; the induced Spin
representation in the latter case doesn’t depend on the chosen representation.
These representations can be used to carry out the following construction:

  1. $(i)$

given a complex Spin representation
$\Delta_{r,1}^{\mathbb{C}}\colon\mathrm{Spin}_{r,1}^{0}\to\text{GL}(V,\mathbb{C})$,
we can consider the _complex spinor bundle_

$S(M)\overset{\cdot}{=}P_{\mathrm{Spin}_{r,1}^{0}}\times_{\Delta_{r,1}^{\mathbb{C}}}V$
(2.6)

  2. $(ii)$

we can also consider the _Clifford algebra bundle_

$\mathscr{C}\ell(M)\overset{\cdot}{=}P_{\text{SO}_{r,1}^{0}}\times_{c\ell}\mathscr{C}\ell_{r,1}$
(2.7)

where $c\ell\colon\text{SO}_{r,1}^{0}\to\text{Aut}(\mathscr{C}\ell_{r,1})$
denotes the unique extension to the Clifford algebra $\mathscr{C}\ell_{r,1}$
of the action of $\text{SO}_{r,1}^{0}$ on $(\mathbb{R}^{r+1},q_{r,1})$. It can
be shown that the fiber of $\mathscr{C}\ell(M)$ at $p\in M$ is isomorphic to
$\mathscr{C}\ell(T_{p}^{\ast}M,g_{p})$.

These two bundles are related by means of the _Clifford module multiplication_

$\mu\colon\mathscr{C}\ell(M)\otimes S(M)\to S(M)$

which explicitly uses the fact that $\Delta_{r,1}^{\mathbb{C}}$ descends from
an algebra representation of $\mathscr{C}\ell_{r,1}$.
Thanks to the spin structure, this map can then be extended to yield a
Clifford module multiplication

$\cdot_{\Gamma}\colon\mathfrak{X}(M)\otimes\Gamma(S(M))\to\Gamma(S(M))$ (2.8)

between vector fields and sections of the spinor bundle.

We shall indicate with

$\nabla^{s}\colon\mathfrak{X}(M)\otimes\Gamma(S(M))\to\Gamma(S(M))$ (2.9)

the covariant derivative induced as of (2.4) on the spinor bundle by the spin
connection $\omega^{s}$. This connection behaves well with respect to the
Clifford multiplcation, in the sense that it is a module derivation: given
$s\in\Gamma(S(M))$ and $v,u\in\mathfrak{X}(M)$ we have

$\nabla_{u}^{s}(v\cdot_{\Gamma}s)=(\nabla_{u}v)\cdot_{\Gamma}s+v\cdot_{\Gamma}\nabla_{u}^{s}s\
.$ (2.10)

The Clifford multiplication can be combined with the covariant derivative
$\nabla^{s}$ and with the musical isomorphism induced by the metric to yield a
partial differential operator acting on sections of the spinor bundle, the
_Dirac operator_

$\begin{split}\mathbf{D}\colon\Gamma(S(M))&\to\Gamma(S(M))\\\
s\qquad&\mapsto\quad\mathbf{D}(s)\overset{\cdot}{=}i\left(\cdot_{\Gamma}(\nabla^{s}(s))\right)\
.\end{split}$ (2.11)

Given a local orthonormal frame $\\{e_{i}\\}\subset\Gamma(TM)$ and its dual
$\\{e^{i}\\}\subset\Gamma(T^{\ast}M)$ and considering the local section
$\sigma\colon U_{\alpha}\to P_{\mathrm{Spin}_{r,1}^{0}}(M)$, (2.11) can be
written locally as

$\mathbf{D}(s)(p)=ie^{j}(p)\cdot_{\Gamma}\left[\sigma(p),e_{j}(p)\left(s^{\sharp}\circ\sigma\right)+{\Delta_{r,1}^{\mathbb{C}}}_{\ast}\left(\omega_{\sigma(p)}^{s}(\sigma_{\ast_{p}}e_{j}(p))\right)s^{\sharp}(\sigma(p))\right]$

where we use Einstein’s convention. As one can see from the local expression
presented above, the Dirac operator is a first-order partial differential
operator whose principal symbol $\sigma(\mathbf{D})$ is given pointwise by

$\sigma(\mathbf{D})\left(p,\xi_{p}\right)(s_{p})=\mu|_{p}(\xi_{p},s_{p})\ .$

Owing to the properties of Clifford multiplication and of the principal
symbol, it is then evident that $\mathbf{D}^{2}$ is a normally hyperbolic,
second-order partial differential operator. Therefore, $\mathbf{D}$ admits
unique advanced and retarded Green operators

$S_{\pm}\colon\Gamma_{c}(S(M))\to\Gamma(S(M))$ (2.12)

i.e. $\mathbf{D}\circ S_{\pm}=\text{id}_{\Gamma(S(M))}$,
$S_{\pm}\circ\mathbf{D}|_{\Gamma_{c}(S(M))}=\text{id}_{\Gamma_{c}(S(M))}$ and
${\mathrm{supp}}(S_{\pm}(s))\subseteq J^{\mp}({\mathrm{supp}}(s))$ [1]. We
remind, for later purposes, that we can also define the so-called _causal
Green operator_ (or, more simply, propagator) by posing $S\doteq
S_{-}-S_{+}$.111We hope the reader shall not feel disheartened by the frequent
abuse of notations, especially here for the symbols of the Green operators.
The context hopefully makes clear which is which.

By [33, Proposition 12.1.65], we know that $S(M)$ admits a Hermitian metric
such that the Clifford multiplication by $\alpha\in\Gamma(T^{\ast}M)$ is an
Hermitian automorphism of $S(M)$. We shall denote the Hermitian metric by
$(\cdot,\cdot)$, and the associated two form by $h$.

###  2.2. The geometric description of charged spinors

Following [42], the coupling of a Dirac field with an external gauge field
exploits the notion of _direct product bundle_ , which we recall here:

######  Definition 2.1.

Let $\pi_{P}\colon P\to M$ and $\pi_{Q}\colon Q\to M$ be two principal $G$\-
and $H$-bundles respectively. The product space $P\times Q$ is a principal
$G\times H$-bundle over $M\times M$; by considering the diagonal
$i\colon\Delta{\hookrightarrow}M\times M$ we can then consider the pullback
bundle

$P+Q\overset{\cdot}{=}i^{\ast}(P\times
Q)=\left\\{(x,p,q)\in\Delta\times(P\times Q)\ \text{s.t.}\
(x,x)=(\pi_{P}(p),\pi_{Q}(q))\right\\}\ .$

As $\Delta$ is diffeomorphic to $M$, we have that $P+Q$ is a principal
$G\times H$-bundle over $M$, called the direct product bundle.

Notice that $P+Q$ can be naturally viewed as a subset of $P\times Q$; we can
consider the restriction of the natural projections

$f_{P}\colon P\times Q\to P\qquad f_{Q}\colon P\times Q\to Q$

to $P+Q$. These are principal bundle homomorphisms, meaning that

$f_{P}\left(r_{(g,h)}(p,q)\right)=r_{g}p\qquad
f_{Q}\left(r_{(g,h)}(p,q)\right)=r_{h}q\ .$

If the two principal bundles in Definition 2.1 are endowed with the
connections $\omega^{P}\in\Omega^{1}(P,\mathfrak{p})$ and
$\omega^{Q}\in\Omega^{1}(Q,\mathfrak{q})$ respectively, then by [29,
Proposition 6.3] we have that the 1-form

$\omega\overset{\cdot}{=}f_{P}^{\ast}\omega^{P}\oplus f_{Q}^{\ast}\omega^{Q}$
(2.13)

is an Ehresmann connection on $P+Q$.
Let us now consider two representations

$\rho_{1}\colon G\to\text{GL}(V)\qquad\rho_{2}\colon H\to\text{GL}(V)$

such that $\rho_{1}(g)\rho_{2}(h)=\rho_{2}(h)\rho_{1}(g)$ for every $g\in G$,
$h\in H$. Then one can construct the representation

$\begin{split}\rho\colon G\times H&\to\text{GL}(V)\\\
(g,h)&\mapsto\rho\left(g,h\right)\overset{\cdot}{=}\rho_{1}(g)\rho_{2}(h)\end{split}$
(2.14)

whose adjoint representation is given by

$\begin{split}\rho_{\ast}\colon\mathfrak{g}\oplus\mathfrak{h}&\to\qquad\text{End}(V)\\\
(\mathbf{g},\mathbf{h})&\mapsto{\rho_{1}}_{\ast}(\mathbf{g})+{\rho_{2}}_{\ast}(\mathbf{h})\end{split}$

and consider the associated bundle

$(P+Q)\times_{\rho}V$

which will then be endowed with a covariant derivative.

In our case of interest we will deal with a principal
$\mathrm{Spin}_{r,1}^{0}$-bundle $\pi_{s}\colon P_{\mathrm{Spin}_{r,1}^{0}}\to
M$ coupled with a principal $G$-bundle $\pi_{g}\colon P_{G}\to M$, where $G$
is a compact Lie group which admits a representation $\rho_{G}\colon
G\to\text{GL}(V)$ such that

$\Delta_{r,1}^{\mathbb{C}}(s)\rho_{G}(g)=\rho_{G}(g)\Delta_{r,1}^{\mathbb{C}}(s)\
\text{for every}\ s\in\mathrm{Spin}_{r,1}^{0},\ g\in G\ .$

By considering the direct product bundle $P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$
endowed with the connection given by (2.13) as well as the representation
given by (2.14) we then can construct the associated vector bundle

$S_{G}(M)\overset{\cdot}{=}(P_{\mathrm{Spin}_{r,1}^{0}}+P_{G})\times_{\rho}V$

which we shall call _charged spinor bundle_. This can be endowed with the
covariant derivative

$\nabla^{s,G}\colon\mathfrak{X}(M)\otimes\Gamma(S_{G}(M))\to\Gamma(S_{G}(M)))$

locally given by

$\begin{split}&(\nabla^{s,G}_{v}s)(p)=\left[\sigma(p),v(p)\left(s^{\sharp_{G}}\circ\sigma\right)+\rho_{\ast}\Big{(}\omega_{\sigma(p)}({\sigma_{\ast}}_{p}v(p))\Big{)}s^{\sharp_{G}}\left(\sigma(p)\right)\right]\\\
&=\left[\sigma(p),v(p)\left(s^{\sharp_{G}}\circ\sigma\right)+\left({\Delta_{r,1}^{\mathbb{C}}}_{\ast}\left((f_{\mathrm{Spin}_{r,1}^{0}}\circ\sigma)^{\ast}\omega^{s}_{p}(v(p))\right)+{\rho_{G}}_{\ast}\Big{(}(f_{G}\circ\sigma)^{\ast}\omega^{G}_{p}(v(p))\Big{)}\right)s^{\sharp_{G}}\left(\sigma(p)\right)\right]\end{split}$

where $\sigma\colon U\to P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$ is a local section
of the direct product bundle and

$\cdot^{\sharp_{G}}\colon\Gamma(\wedge^{q}T^{\ast}M\otimes
S_{G}(M))\to\Omega^{q}_{\rho}(P_{\mathrm{Spin}_{r,1}^{0}}+P_{G},V)$

denotes the isomorphism as of (2.2). Notice that as $V$ is a
$\mathscr{C}\ell_{r,1}$-module, even in this case we have a Clifford module
multiplication

$\cdot_{\Gamma,G}\colon\mathfrak{X}(M)\otimes\Gamma(S_{G}(M))\to\Gamma(S_{G}(M))$

which we can use to construct a Dirac operator $\mathbf{D}^{G}$ as in (2.11).
Notice that the highest-order term is analogous to that of $\mathbf{D}$,
therefore the principal symbols of these two operators coincide: it follows
that $\mathbf{D}^{G}$ admits unique advanced and retarded Green operators

$S^{G}_{\pm}\colon\Gamma_{c}(S_{G}(M))\to\Gamma(S_{G}(M))\ ,$ (2.15)

with the analogous properties of the uncharged operators as after (2.12).
Again, we can define the _causal Green operator_ by posing $S^{G}\doteq
S^{G}_{-}-S^{G}_{+}$.

##  3\. The classical Møller map on field configurations

###  3.1. The entwining maps between sections

In order to compare the two theories, we need a way to relate two different
spaces of functions: on one hand we have the smooth sections of the spinor
bundle $\Gamma(S(M))$, while on the other we have the sections of the charged
spinor bundle $\Gamma(S_{G}(M))$. Notice that these spaces are isomorphic,
thanks to the linear isomorphisms in (2.2), to the spaces

$\begin{gathered}F\overset{\cdot}{=}\left\\{f\colon
P_{\mathrm{Spin}_{r,1}^{0}}\to V,\ f\ \text{right}-\mathrm{Spin}_{r,1}^{0}\
\text{equivariant}\right\\}\\\ F_{G}\overset{\cdot}{=}\left\\{f\colon
P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}\to V,\ f\
\text{right}-\mathrm{Spin}_{r,1}^{0}\times G\
\text{equivariant}\right\\}\end{gathered}$

respectively. In this first instance, we shall work with trivial principal
bundles. Notice that owing to [22, Section I] principal
$\mathrm{Spin}_{r,1}^{0}$-bundles over non-compact spacetimes admit a global
section $\sigma\colon M\to P_{\mathrm{Spin}_{r,1}^{0}}$, and are thus always
trivial. To simplify the computations and the definition in this first case,
we make the following assumptions:

  1. (i)

_we assume that the principal $G$-bundle $P_{G}$ is trivial_; this holds, for
instance, if we consider a contractible spacetime, or if we consider principal
bundles $P_{U(1)}$ whose first characteristic class $c_{1}(P)$ vanishes.

As $P_{G}$ is trivial, there exists a global section $\sigma_{G}\colon M\to
P_{G}$; we can use it to define the global section

$M\ni p\overset{\tilde{\sigma}}{\mapsto}(p,\sigma(p),\sigma_{G}(p))$

of the direct product bundle $P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$. It follows
easily that $f_{\mathrm{Spin}_{r,1}^{0}}\circ\tilde{\sigma}=\sigma$,
$f_{G}\circ\tilde{\sigma}=\sigma_{G}$. Moreover, notice that as the principal
bundles $P_{\mathrm{Spin}_{r,1}^{0}}$ and $P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$
are trivial, the associated fiber bundles are also trivial, with
diffeomorphisms given by

$\displaystyle\varphi^{s}\colon(M\times\mathrm{Spin}_{r,1}^{0})\times_{\Delta_{r,1}^{\mathbb{C}}}V$
$\displaystyle\to\quad M\times
V\qquad\varphi^{s,G}\colon(M\times(\mathrm{Spin}_{r,1}^{0}\times
G))\times_{\rho}V$ $\displaystyle\to\quad M\times V$
$\displaystyle[(p,s),v]\qquad$
$\displaystyle\mapsto(p,\Delta_{r,1}^{\mathbb{C}}(s)v)\qquad\qquad\qquad[(p,s,g),v]$
$\displaystyle\mapsto(p,\rho(s,g)v)\ .$

As the vector bundles are trivial, the Hermitian metrics $(\cdot,\cdot)$ on
$S(M)$ and $(\cdot,\cdot)_{G}$ on $S_{G}(M)$ can be induced by a Hermitian
metric $(\cdot,\cdot)_{V}$ on $V$; notice that thanks to [33, Proposition
12.1.27] the Hermitian metric can be chosen to be such that Clifford
multiplication satisfies

$\rho(\phi)^{\ast}=-\rho(\phi^{\dagger}),\qquad\phi^{\dagger}=\tilde{\alpha}(\phi^{t})$

where $\cdot^{t}\colon\mathscr{C}\ell_{r,1}\to\mathscr{C}\ell_{r,1}$ denotes
the transpose and $\tilde{\alpha}$ is the extension of
$\mathbb{R}^{r,1}\ni\mathbf{v}\to-\mathbf{v}$ to $\mathscr{C}\ell_{r,1}$. We
shall denote the Hermitian matrix associated with $(\cdot,\cdot)_{V}$ by
$(h_{ij})$.

  2. (ii)

As a second core assumption, _we shall assume that the (global) gauge
potential_

$\mathscr{A}(\sigma,\omega)\doteq\sigma_{G}^{\ast}\omega^{G}=(f_{G}\circ\tilde{\sigma})^{\ast}\omega^{G}\in\Omega^{1}(M,\mathfrak{g})$

_is compactly supported_ ; its support will be denoted by
$\mathrm{supp}(\mathscr{A})$.

  3. (iii)

_We will assume that the representation $\rho_{G}\colon G\to\mathrm{GL}(V)$
commutes with the Clifford algebra representation inducing
$\Delta_{r,1}^{\mathbb{C}}\colon\mathrm{Spin}_{r,1}^{0}\to\mathrm{GL}(V)$_.
Notice that this greatly reduces the freedom in the group $G$: indeed, it can
be shown [30, Theorem 4.3] that

$\begin{split}\mathbb{C}\ell_{r+1}\simeq
M(2^{\lfloor(r+1)/2\rfloor},\mathbb{C})&\ \text{if}\ r+1=0\mod 2\\\
\mathbb{C}\ell_{r+1}\simeq M(2^{\lfloor(r+1)/2\rfloor},\mathbb{C})\oplus
M(2^{\lfloor(r+1)/2\rfloor},\mathbb{C})&\ \text{if}\ r+1=1\mod
2\qquad\end{split}$

and that an irreducible $\mathbb{C}$-module for $\mathscr{C}\ell_{r,1}$ (which
descends from an irreducible module for $\mathbb{C}\ell_{r+1}$) has complex
dimension $2^{\lfloor(r+1)/2\rfloor}$. We are therefore assuming that the
image of $\rho_{G}$ lies in the centre of the algebra
$M(2^{\lfloor(r+1)/2\rfloor},\mathbb{C})$, that is,
$\rho_{G}(G)\subseteq\mathbb{C}\mathrm{id}_{V}$. Although it may seem rather
restrictive, this case encompasses the interesting case of $G=U(1)$, that is,
the case of electrically charged spinors.

The global section $\sigma_{G}\colon M\to P_{G}$ allows us to construct the
following maps:

######  Definition 3.1.

Let us define the maps $\mathcal{p}\colon F_{G}\to F$,

$F_{G}\ni
f_{g}\mapsto(\mathcal{p}f_{g})(p,s)\overset{\cdot}{=}f_{g}(p,s,\sigma_{G}(p))$
(3.1)

and $\mathcal{i}\colon F\to F_{G}$,

$F\ni
f\mapsto(\mathcal{i}f)(p,s,g)\overset{\cdot}{=}\rho_{G}(\tilde{g}^{-1})f(p,s)$
(3.2)

where $\tilde{g}\in G$ is the unique group element such that
$r_{\tilde{g}}\sigma_{G}(p)=(p,g)$, which exists as the action of $G$ on
$P_{G}$ is free and transitive.

Notice that Definition 3.1 is well-posed, in the sense that given $f_{g}\in
F_{G}$, $\mathcal{p}f_{g}$ is $\text{right}-\mathrm{Spin}_{r,1}^{0}$
equivariant, and given $f\in F$, $\mathcal{i}f$ is
$\text{right}-\mathrm{Spin}_{r,1}^{0}\times G$ equivariant: indeed,

$\begin{split}\left(r_{\overline{s}}^{\ast}(\mathcal{p}f_{g})\right)(p,s)&=(\mathcal{p}f_{g})(p,r_{\overline{s}}s)=f_{g}(p,r_{\overline{s}}s,\sigma_{G}(p))=(r_{(\overline{s},\text{id}_{G})}^{\ast}f_{g})(p,s,\sigma_{G}(p))\\\
&=\rho(\overline{s}^{-1},\text{id}_{G})f_{g}(p,s,\sigma_{G}(p))=\Delta_{r,1}^{\mathbb{C}}(\overline{s}^{-1})(\mathcal{p}f_{g})(p,s)\end{split}$

and

$\begin{split}\left(r_{(\overline{s},\overline{g})}^{\ast}(\mathcal{i}f)\right)(p,s,g)&=(\mathcal{i}f)(p,r_{\overline{s}}s,r_{\overline{g}}g)=\rho_{G}(\overline{g}^{-1})\rho_{G}(\tilde{g}^{-1})f(p,r_{\overline{s}}s)\\\
&=\rho_{G}(\overline{g}^{-1})\rho_{G}(\tilde{g}^{-1})r_{\overline{s}}^{\ast}f(p,s)=\rho_{G}(\overline{g}^{-1})\rho_{G}(\tilde{g}^{-1})\Delta_{r,1}^{\mathbb{C}}(\overline{s}^{-1})f(p,s)\\\
&=\Delta_{r,1}^{\mathbb{C}}(\overline{s}^{-1})\rho_{G}(\overline{g}^{-1})\rho_{G}(\tilde{g}^{-1})f(p,s)=\rho(\overline{s}^{-1},\overline{g}^{-1})(\mathcal{i}f)(p,s,g)\
.\end{split}$

We can combine the maps $\mathcal{i}$ and $\mathcal{p}$ defined in Definition
3.1 with the linear maps in (2.2) to yield maps

$\begin{split}\mathcal{i}\colon\Gamma(S(M))\to\Gamma(S_{G}(M))&\qquad\mathcal{p}\colon\Gamma(S_{G}(M))\to\Gamma(S(M))\\\
s\hskip 14.22636pt\mapsto\hskip
8.5359pt(\mathcal{i}s^{\sharp})^{\flat_{G}}\hskip 8.5359pt&\qquad\hskip
31.2982pts\hskip 14.22636pt\mapsto\hskip
8.5359pt(\mathcal{p}s^{\sharp_{G}})^{\flat}\end{split}$ (3.3)

which are denoted with the same symbol for the sake of notational simplicity.
Let us now endow the spaces $\Gamma(S(M))$ and $\Gamma(S_{G}(M))$ with the
Fréchet topology induced by the families of seminorms

$\begin{split}\left\lVert s\right\rVert_{K,n}&\overset{\cdot}{=}\max_{0\leq
i\leq n}{\sup_{p\in
K}\left\lVert\left({\nabla^{s}}^{i}s\right)(p)\right\rVert_{S(M)\otimes
T^{\ast}M^{\otimes i}}}\\\ \left\lVert
s\right\rVert_{K,n}^{G}&\overset{\cdot}{=}\max_{0\leq i\leq n}{\sup_{p\in
K}\left\lVert\left({\nabla^{s,G}}^{i}s\right)(p)\right\rVert_{S_{G}(M)\otimes
T^{\ast}M^{\otimes i}}}\ .\end{split}$ (3.4)

We shall indicate these topological spaces as $\pazocal{E}(S(M))$ and
$\pazocal{E}(S_{G}(M))$ respectively.

######  Lemma 3.2.

The maps $\mathcal{i}$ and $\mathcal{p}$ defined in (3.3) are continuous with
respect to the Fréchet topologies on $\pazocal{E}(S(M))$ and
$\pazocal{E}(S_{G}(M))$.

###### Proof.

We need to exhibit, for each $K\subset\subset M$ and for each
$n\in\mathbb{N}$, constants $k_{K,n},k_{K,n}^{G}\in\mathbb{R}^{+}$ and
families $\\{(K_{l},n_{l})\\}_{1\leq l\leq
m},\\{(K_{l}^{G},n_{l}^{G})\\}_{1\leq l\leq m^{G}}$ such that

$\left\lVert\mathcal{i}s\right\rVert^{G}_{K,n}\leq k_{K,n}\max_{1\leq l\leq
m}\left\lVert
s\right\rVert_{K_{l},n_{l}}\qquad\left\lVert\mathcal{p}s\right\rVert_{K,n}\leq
k_{K,n}^{G}\max_{1\leq l\leq m^{G}}\left\lVert
s\right\rVert^{G}_{K_{l}^{G},n_{l}^{G}}\ .$

The case $n=0$ is easy: indeed,

$\begin{split}\left\lVert\mathcal{i}s\right\rVert^{G}_{K,0}&={\sup_{p\in
K}}\left\lVert(\mathcal{i}s)(p)\right\rVert_{S_{G}(M)}=\sup_{p\in
K}\left\lVert\rho(\pi_{2}(\tilde{\sigma}(p)))(\mathcal{i}s)^{\sharp_{G}}(\tilde{\sigma}(p))\right\rVert_{V}\\\
&=\sup_{p\in
K}\left\lVert\rho_{G}(\pi_{2}\circ\sigma_{G}(p))\Delta_{r,1}^{\mathbb{C}}(\pi_{2}\circ\sigma(p))s^{\sharp}(\sigma(p))\right\rVert_{V}\\\
&\leq\sup_{p\in
K}\left\lVert\rho_{G}\left(\pi_{2}\circ{\sigma_{G}(p)}\right)\right\rVert_{\text{End}(V)}\sup_{p\in
K}\left\lVert\Delta_{r,1}^{\mathbb{C}}(\pi_{2}\circ\sigma(p))s^{\sharp}(\sigma(p))\right\rVert_{V}=k_{K,0}\left\lVert
s\right\rVert_{K,0}\end{split}$

where we use that $K\subset\subset M$ and that $\rho_{G}$ is a smooth
representations onto a matrix algebra on $V$ finite-dimensional vector space.
Consider now $n=1$; in this case we compute
$\nabla_{e_{i}}^{s,G}(\mathcal{i}s)(p)$, where $\\{e_{i}\\}_{1\leq
i\leq\dim(M)}$ is a local pesudo-orthonormal basis for $TM$. In particular,
using a section $\widehat{\sigma}\colon M\to
P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$ with
$\widehat{\sigma}=r_{g}\tilde{\sigma}$,

$\begin{split}\nabla_{e_{i}}^{s,G}(\mathcal{i}s)(p)&=\Big{[}\widehat{\sigma}(p),e_{i}(p)\Big{(}\rho_{G}\left(\pi_{G}\circ
g(p)\right)^{-1}s^{\sharp}(f_{P}\circ\widehat{\sigma}(p))\Big{)}+\Big{(}{\Delta_{r,1}^{\mathbb{C}}}_{\ast}\left(\left((f_{P}\circ\widehat{\sigma})^{\ast}\omega^{s}\right)_{p}\left(e_{i}(p)\right)\right)\\\
&\quad+{\rho_{G}}_{\ast}\left(\left((f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}\left(e_{i}(p)\right)\right)\Big{)}(\mathcal{i}s)^{\sharp_{G}}(\widehat{\sigma}(p))\Big{]}\\\
&=\left[\widehat{\sigma}(p),\rho_{G}\left(\pi_{G}\circ
g(p)\right)^{-1}\left(e_{i}(p)\left(s^{\sharp}\circ
f_{P}\circ\widehat{\sigma}\right)+{\Delta_{r,1}^{\mathbb{C}}}_{\ast}\left(\left((f_{P}\circ\widehat{\sigma})^{\ast}\omega^{s}\right)_{p}(e_{i}(p))\right)s^{\sharp}(\sigma(p))\right)\right]\\\
&\quad+\left[\widehat{\sigma}(p),e_{i}(p)\left(\rho_{G}\left(\pi_{G}\circ
g(p)\right)^{-1}\right)s^{\sharp}(f_{P}\circ\widehat{\sigma}(p))+{\rho_{G}}_{\ast}\left(\left((f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)(is)^{\sharp_{G}}(\widehat{\sigma}(p))\right]\
.\end{split}$

Notice that the first term is equal to $\mathcal{i}\circ\nabla^{s}_{e_{i}}s$,
while the second one can be written as

$\left[\widehat{\sigma}(p),{\rho_{G}}_{\ast}\left(\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)(\mathcal{i}s)^{\sharp_{G}}(\widehat{\sigma}(p))\right]$

where $\theta_{G}$ denots the Maurer-Cartan form of $G$. Using the notation
$\varepsilon_{i}=g(e_{i},e_{i})$ we have

$\begin{split}&\left\lVert\nabla^{s,G}(\mathcal{i}s)(p)\right\rVert_{S_{G}(M)\otimes
T^{\ast}M}^{2}=\sum_{i=1}^{\dim
M}\varepsilon_{i}\left\lVert\nabla^{s,G}_{e_{i}}(\mathcal{i}s)(p)\right\rVert_{S_{G}(M)}^{2}\\\
&\leq k\sum_{i=1}^{\dim
M}\varepsilon_{i}\left\lVert\mathcal{i}\circ\nabla^{s}_{e_{i}}s(p)\right\rVert_{S_{G}(M)}^{2}\\\
&\qquad+k\sum_{i=1}^{\dim
M}\varepsilon_{i}\left\lVert\rho\left(\pi_{2}\circ\widehat{\sigma}(p)\right){\rho_{G}}_{\ast}\Big{(}\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\Big{)}(\mathcal{i}s)^{\sharp_{G}}(\sigma(p))\right\rVert_{V}^{2}\
.\end{split}$

Given a compact set $K\subset\subset M$, the first term can be bounded from
above by

$k_{1}\sup_{p\in K}\left\lVert\nabla^{s}s\right\rVert_{S(M)\otimes
T^{\ast}M}^{2}\leq k_{1}\left\lVert s\right\rVert_{K,1}^{2}$

while for the second term we have the upper bound

$k_{2}\sup_{p\in K}\left\lVert s\right\rVert_{S(M)}^{2}=k_{2}\left\lVert
s\right\rVert_{K,0}^{2}\leq k_{2}\left\lVert s\right\rVert_{K,1}^{2}\ .$

Therefore we can give the estimate

$\left\lVert\mathcal{i}s\right\rVert^{G}_{K,1}\leq k_{K,1}\left\lVert
s\right\rVert_{K,1}\ .$

We proceed by induction: suppose we have an inequality of the form
$\left\lVert\mathcal{i}s\right\rVert^{G}_{K,i}\leq k_{K,i}\left\lVert
s\right\rVert_{K,i}$ for $i$ up to $n-1\in\mathbb{N}$. Then by the above
discussion ${\nabla^{s,G}}^{n}(\mathcal{i}s)$ can be written as
$\mathcal{i}{\nabla^{s}}^{n}s$, which can be bounded by $k_{n}\left\lVert
s\right\rVert_{K,n}$, plus lower order terms in the covariant derivative of
$s$ and of the map ${\rho_{G}}_{\ast}\left(\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)$
which are bounded in every compact set $K\subset\subset M$ by the inductive
hypothesis and the smoothness of the involved maps respectively.

An analogous proof holds for the map
$\mathcal{p}\colon\Gamma(S_{G}(M))\to\Gamma(S(M))$: indeed,

$\begin{split}\left\lVert\mathcal{p}s\right\rVert_{K,0}&=\sup_{p\in
K}\left\lVert(\mathcal{p}s)(p)\right\rVert_{S(M)}=\sup_{p\in
K}\left\lVert\Delta_{r,1}^{\mathbb{C}}(\pi_{2}\circ\sigma(p))(\mathcal{p}s)^{\sharp}\left(\sigma(p)\right)\right\rVert_{V}=\sup_{p\in
K}\left\lVert\Delta_{r,1}^{\mathbb{C}}(\pi_{2}\circ\sigma(p))s^{\sharp_{G}}\left(\tilde{\sigma}(p)\right)\right\rVert_{V}\\\
&\leq\sup_{p\in
K}\left\lVert\rho_{G}(\pi_{2}\circ\sigma_{G}(p))^{-1}\right\rVert_{\text{End}(V)}\sup_{p\in
K}\left\lVert\rho_{G}(\pi_{2}\circ\sigma_{G}(p))\Delta_{r,1}^{\mathbb{C}}(\pi_{2}\circ\sigma(p))s^{\sharp_{G}}\left(\tilde{\sigma}(p)\right)\right\rVert_{V}\\\
&=k^{G}_{K,0}\sup_{p\in K}\left\lVert s\right\rVert_{S^{G}(M)}\end{split}$

while for the case $n=1$ we have

$\begin{split}\nabla^{s}_{e_{i}}(\mathcal{p}s)(p)&=\left[\sigma(p),e_{i}(p)\left(s^{\sharp_{G}}(\tilde{\sigma}(p))\right)+{\Delta_{r,1}^{\mathbb{C}}}_{\ast}\Big{(}(\sigma^{\ast}\omega^{s})_{p}(e_{i}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\right]\\\
&=\Big{[}\sigma(p),e_{i}(p)\left(s^{\sharp_{G}}(\tilde{\sigma}(p))\right)+{\Delta_{r,1}^{\mathbb{C}}}_{\ast}\Big{(}(\sigma^{\ast}\omega^{s})_{p}(e_{i}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\\\
&\quad+{\rho_{G}}_{\ast}\Big{(}(\sigma_{G}^{\ast}\omega^{G})_{p}(e_{i}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))-{\rho_{G}}_{\ast}\Big{(}(\sigma_{G}^{\ast}\omega^{G})_{p}(e_{i}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\Big{]}\
.\end{split}$

Notice that

$\left[\sigma(p),e_{i}(p)(s^{\sharp_{G}}\circ\tilde{\sigma})+\left({\Delta_{r,1}^{\mathbb{C}}}_{\ast}\Big{(}(\sigma^{\ast}\omega^{s})_{p}(e_{i}(p))\Big{)}+{\rho_{G}}_{\ast}\Big{(}(\sigma_{G}^{\ast}\omega^{G})_{p}(e_{i}(p))\Big{)}\right)s^{\sharp_{G}}(\tilde{\sigma}(p))\right]=\mathcal{p}\left(\nabla_{e_{i}}^{s,G}s\right)(p)\
.$

Then

$\begin{split}&\left\lVert\nabla^{s}(\mathcal{p}s)(p)\right\rVert_{S(M)\otimes
T^{\ast}M}^{2}=\sum_{i=1}^{\dim
M}\varepsilon_{i}\left\lVert\nabla^{s}_{e_{i}}(\mathcal{p}s)(p)\right\rVert_{S(M)}^{2}\\\
&\leq k\sum_{i=1}^{\dim
M}\varepsilon_{i}\left\lVert\mathcal{p}\left(\nabla_{e_{i}}^{s,G}s\right)(p)\right\rVert^{2}_{S(M)}+k\sum_{i=1}^{\dim
M}\varepsilon_{i}\left\lVert\Delta_{r,1}^{\mathbb{C}}\left(\pi_{2}\circ\sigma(p)\right){\rho_{G}}_{\ast}\Big{(}(\sigma_{G}^{\ast}\omega^{G})_{p}(e_{i}(p)))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\right\rVert_{V}^{2}\
.\end{split}$

As before, given a compact $K\subset\subset M$ the first term can be bounded
from above by

$k_{1}\sup_{p\in K}\left\lVert\nabla^{s,G}s\right\rVert_{S_{G}(M)\otimes
T^{\ast}M}\leq k_{1}\left({\left\lVert s\right\rVert^{G}_{K,1}}\right)^{2}$

and the second term can be bounded from above by

$k_{2}\sup_{p\in K}\left\lVert s\right\rVert_{S_{G}(M)}^{2}\leq
k_{2}\left({\left\lVert s\right\rVert^{G}_{K,0}}\right)^{2}\ .$

By reasoning as above by induction we have the desired result. ∎

###  3.2. The explicit construction of the classical Møller map

We now use these maps, in particular
$\mathcal{i}\colon\pazocal{E}(S(M))\to\pazocal{E}(S_{G}(M))$ to give an
explicit formula for the classical Møller operator, whose properties in the
examined case we recall:

######  Definition 3.3.

The _classical Møller map on field configurations_ is a map
$R_{A}\colon\pazocal{E}(S(M))\to\pazocal{E}(S_{G}(M))$ such that

  1. $(i)$

$\mathbf{D}^{G}\circ R_{A}=\mathcal{i}\circ\mathbf{D}$ ,

  2. $(ii)$

$R_{A}(s)|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}=(\mathcal{i}s)|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}$ .

In this case, as anticipated, the classical Møller operator has an explicit
form:

######  Theorem 3.4.

The unique solution to the requirements in Definition 3.3 is given by

$R_{A}=\mathcal{i}-S^{G}_{-}\circ A\circ\mathcal{i}$ (3.5)

where $A\colon\pazocal{E}(S_{G}(M))\to\pazocal{E}(S_{G}(M))$ acts pointwise as

$(As)(p)\overset{\cdot}{=}ie^{j}(p)\cdot_{\Gamma,G}\left[\widehat{\sigma}(p),{\rho_{G}}_{\ast}\Big{(}\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{j}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\right]$
(3.6)

with $g\colon M\to\mathrm{Spin}_{r,1}^{0}\times G$ such that
$\widehat{\sigma}=r_{g}\tilde{\sigma}$.

######  Remark 3.5.

Notice that the map $A\colon\pazocal{E}(S_{G}(M))\to\pazocal{E}(S_{G}(M))$, as
it is written in (3.6), is gauge independent: indeed, given another section
$\overline{\sigma}\colon M\to P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$ we have that
$\overline{\sigma}=r_{g^{\prime}}\circ\widehat{\sigma}$ where
$g^{\prime}\colon M\to\mathrm{Spin}_{r,1}^{0}\times G$ and

$f_{P}(r_{g^{\prime}}\circ\widehat{\sigma})=r_{\pi_{\mathrm{Spin}}(g^{\prime})}f_{P}(\widehat{\sigma})\qquad
f_{G}(r_{g^{\prime}}\circ\widehat{\sigma})=r_{\pi_{G}(g^{\prime})}f_{G}(\widehat{\sigma})\
.$

The following relations holds between the different pullbacks of the
connection form $\omega^{G}$ and of the Maurer-Cartan form $\theta_{G}$:

$\begin{split}(f_{G}\circ\overline{\sigma})^{\ast}\omega^{G}&=(r_{\pi_{G}(g^{\prime})}\circ
f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}=\text{Ad}_{\pi_{G}(g^{\prime})^{-1}}\circ(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}+(\pi_{G}(g^{\prime}))^{\ast}\theta_{G}\
,\\\
(\pi_{G}(gg^{\prime}))^{\ast}\theta_{G}&=(r_{\pi_{G}(g^{\prime})}\circ\pi_{G}(g))^{\ast}\theta_{G}=\text{Ad}_{\pi_{G}(g^{\prime})^{-1}}\circ(\pi_{G}(g))^{\ast}\theta_{G}+(\pi_{G}(g^{\prime}))^{\ast}\theta_{G}\
.\end{split}$

Therefore,

$\begin{split}&ie^{j}(p)\cdot_{\Gamma,G}\left[\overline{\sigma}(p),{\rho_{G}}_{\ast}\Big{(}\left(-(\pi_{G}(gg^{\prime}))^{\ast}\theta_{G}+(f_{G}\circ\overline{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{j}(p))\Big{)}s^{\sharp_{G}}(\overline{\sigma}(p))\right]\\\
&=ie^{j}(p)\cdot_{\Gamma,G}\left[\widehat{\sigma}(p),\rho(g^{\prime}(p)){\rho_{G}}_{\ast}\Big{(}\text{Ad}_{\pi_{G}(g^{\prime})^{-1}}\circ\left((f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}-(\pi_{G}(g))^{\ast}\theta_{G}\right)\Big{)}_{p}(e_{j}(p))\rho({g^{\prime}}^{-1}(p))s^{\sharp_{G}}(\widehat{\sigma}(p))\right]\\\
&=ie^{j}(p)\cdot_{\Gamma,G}\left[\widehat{\sigma}(p),{\rho_{G}}_{\ast}\Big{(}\left(-(\pi_{G}(g))^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{j}(p))\Big{)}s^{\sharp_{G}}(\widehat{\sigma}(p))\right]\
.\end{split}$

In our case we do have a preferred gauge choice, given by the section
$\tilde{\sigma}\colon M\to P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$; using that
section, as $g\equiv\text{id}_{G}$ we then have

$(As)(p)=ie^{j}(p)\cdot_{\Gamma,G}\left[\tilde{\sigma}(p),{\rho_{G}}_{\ast}\Big{(}\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{j}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\right]$

which, by the properties required to $\sigma_{G}^{\ast}\omega^{G}$, is
compactly supported in $\mathrm{supp}(\mathscr{A})$. As such, in the following
the gauge potential shall be the one induced by the section ${\sigma}_{G}$.

###### Proof of Theorem 3.4.

First of all, notice that the composition $S_{-}^{G}\circ A\circ\mathcal{i}$
is well-defined thanks to the support properties of
$\sigma_{G}^{\ast}\omega^{G}$: indeed, we know that

$S_{-}^{G}\colon\pazocal{D}(S_{G}(M))\to\pazocal{E}(S_{G}(M))$

and as $As$ vanishes outside of $\mathrm{supp}(\mathscr{A})$ for every
$s\in\pazocal{E}({S}_{G}(M))$ we have that
$(A\circ\mathcal{i})(s)\in\pazocal{D}(S_{G}(M))$ for every
$s\in\pazocal{E}(S(M))$.
Thanks to the properties of the retarded propagator $S_{-}^{G}$, we thus have
that

${\mathrm{supp}}\left((S_{-}^{G}\circ A\circ\mathcal{i})(s)\right)\subseteq
J^{+}\left({\mathrm{supp}}(A\circ\mathcal{i})(s)\right)\subseteq
J^{+}\left(\mathrm{supp}(\mathscr{A})\right)$

and therefore the requirement (ii) in Definition 3.3 is fulfilled.

To proceed further, we need to show that
$\mathcal{i}\colon\pazocal{E}(S(M))\to\pazocal{E}(S_{G}(M))$ commutes with the
Clifford multiplication by sections of the cotangent bundle. That is, we need
to show that

$e^{i}\cdot_{\Gamma,G}\mathcal{i}(u)=\mathcal{i}\left(e^{i}\cdot_{\Gamma}u\right)\
.$

We do so by investigating the result locally. Given a section
$\widehat{\sigma}\colon M\to P_{\mathrm{Spin}_{r,1}^{0}}+P_{G}$ inducing a
section $f_{P}\circ\widehat{\sigma}\colon M\to P_{\mathrm{Spin}_{r,1}^{0}}$,
we have that locally
$\mathcal{i}(u(p))=[\widehat{\sigma}(p),\rho_{G}((\pi_{2}\circ
h)(p))^{-1}u^{\sharp}(f_{P}\circ\widehat{\sigma}(p))]$ where $h\colon
M\to\mathrm{Spin}_{r,1}^{0}\times G$ is such that
$\widehat{\sigma}=r_{h}\tilde{\sigma}$. Then

$\begin{split}e^{i}(p)\cdot_{\Gamma,G}\mathcal{i}(u)(p)&=\left[\widehat{\sigma}(p),\left(g^{ij}(p)e_{j}(p)\right)\cdot_{V}\rho_{G}\left((\pi_{2}\circ
h)(p)\right)^{-1}u^{\sharp}(f_{P}\circ\widehat{\sigma}(p))\right]\\\
&=\left[\widehat{\sigma}(p),\rho_{G}\left((\pi_{2}\circ
h)(p)\right)^{-1}\left(\left(g^{ij}(p)e_{j}(p)\right)\cdot_{V}u^{\sharp}(f_{P}\circ\widehat{\sigma}(p))\right)\right]\\\
&=\mathcal{i}(e^{i}\cdot_{\Gamma}u)(p)\ .\end{split}$

Thus, if we consider now $\mathbf{D}^{G}\circ(\mathcal{i}-S_{-}^{G}\circ
A\circ\mathcal{i})$ by the proof of Lemma 3.2 and by the above discussion we
know that

$\mathbf{D}^{G}\circ\mathcal{i}=\mathcal{i}\circ\mathbf{D}+A\circ\mathcal{i}$

and using the property $\mathbf{D}^{G}\circ
S_{-}^{G}=\text{id}_{\pazocal{D}(S_{G}(M))}$ of the retarded propagator
associated to $\mathbf{D}^{G}$ we have $\mathbf{D}^{G}\circ(S_{-}^{G}\circ
A\circ\mathcal{i})=A\circ\mathcal{i}$; therefore

$\mathbf{D}^{G}\circ(\mathcal{i}-S_{-}^{G}\circ
A\circ\mathcal{i})=\mathcal{i}\circ\mathbf{D}\ .$

Therefore, also the requirement (i) in Definition 3.3 is fulfilled. Thus we
can say that

$R_{A}=\mathcal{i}-S_{-}^{G}\circ A\circ\mathcal{i}\ .$

As far as the uniqueness statement is concerned, we proceed as done in [14]
and [23]: given $s\in\pazocal{E}(S(M))$, let us define
$R_{A}s\overset{\cdot}{=}\psi\in\pazocal{E}(S_{G}(M))$; then we have that
$\psi$ satisfies the following:

$\mathbf{D}^{G}\psi=(\mathcal{i}\circ\mathbf{D})s\ ,\quad\psi|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}=(\mathcal{i}s)|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}\ .$

As $M$ is globally hyperbolic, it is isometric to $\mathbb{R}\times\Sigma$,
with $\\{a\\}\times\Sigma$ Cauchy surface for every $a\in\mathbb{R}$. In
particular, we can assume that $\\{0\\}\times\Sigma\subseteq M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))$. Let $\\{K_{n}\\}_{n\in\mathbb{N}}$ be an
invading sequence of compact sets for $\\{0\\}\times\Sigma$, and define

$\widehat{K}_{n}\overset{\cdot}{=}D(K_{n})\cap[-n,n]\times\Sigma$

where $D(K_{n})$ denotes the Cauchy development of $K_{n}$. We then consider
the family $\\{\chi_{n}\\}_{n\in\mathbb{N}}\subseteq\pazocal{D}(M)$, with
$\chi_{n}\equiv 1$ on $\widehat{K}_{n}$. Using these family, we consider

$\begin{cases}(\mathbf{D}^{G}\circ\mathbf{D}^{G})\phi_{n}=(\mathcal{i}\circ\mathbf{D})(\chi_{n}s)\,&\text{on}\
M\\\ \phi_{n}=(S_{-}^{G}\circ\mathcal{i})(\chi_{n}s)\,&\text{on}\
\\{0\\}\times\Sigma\\\
\nabla^{s,G}_{\nu}\phi_{n}=\nabla^{s,G}_{\nu}\left((S_{-}^{G}\circ\mathcal{i})(\chi_{n}s)\right)\,&\text{on}\
\\{0\\}\times\Sigma\ .\end{cases}$

Notice that $\mathbf{D}^{G}\circ\mathbf{D}^{G}$ is normally hyperbolic, and
thus the above system admits a unique solution which depends continuously on
the initial data [3]. Notice in particular that on $M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))$ the map

$(S_{-}^{G}\circ\mathcal{i})(\chi_{n}s)$

is a solution of the above problem, as there the part due to the gauge
potential vanishes. Therefore by the uniqueness of the solution we have

$\phi_{n}|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}=(S_{-}^{G}\circ\mathcal{i})(\chi_{n}s)|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}\ .$

By the properties of $\\{\widehat{K}_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\chi_{n}\\}_{n\in\mathbb{N}}$ and an analogous reasoning, we have that if
$m>n$ then $\phi_{m}=\phi_{n}$ on $\widehat{K}_{n}$; therefore by defining

$\phi(p)\overset{\cdot}{=}\phi_{n}(p)\ \text{with}\ n\ \text{such that}\
p\in\widehat{K}_{n}$

and $\psi\overset{\cdot}{=}\mathbf{D}^{G}\phi$ we have that

$\mathbf{D}^{G}\psi=(\mathbf{D}^{G}\circ\mathbf{D}^{G})\phi=(\mathcal{i}\circ\mathbf{D})s\
,\quad\psi|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}=(\mathbf{D}^{G}\circ\phi)|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}=(\mathcal{i}s)|_{M\setminus
J^{+}(\mathrm{supp}(\mathscr{A}))}\ .$

By the uniqueness, the $\psi$ above is unique for every $s$, and therefore
$R_{A}$ is as well. ∎

######  Remark 3.6.

Notice that $R_{A}\colon\pazocal{E}(S(M))\to\pazocal{E}(S_{G}(M))$ admits both
a right and left inverse. Let us consider the map

$\widehat{R}_{A}\overset{\cdot}{=}\mathcal{p}+S_{-}\circ\mathcal{p}\circ A$
(3.7)

and consider

$R_{A}\circ\widehat{R}_{A}=\mathcal{i}\circ\mathcal{p}+\mathcal{i}\circ
S_{-}\circ\mathcal{p}\circ A-S_{-}^{G}\circ
A\circ\mathcal{i}\circ\mathcal{p}-S_{-}^{G}\circ A\circ\mathcal{i}\circ
S_{-}\circ\mathcal{p}\circ A\ .$

It is easy to see that $\mathcal{i}\circ\mathcal{p}=\text{id}$ and
$\mathcal{p}\circ\mathcal{i}=\text{id}$ on ${\pazocal{E}(S_{G}(M))}$ and
${\pazocal{E}(S(M))}$ respectively, while by the previous proof we know that
$A\circ\mathcal{i}=\mathbf{D}^{G}\circ\mathcal{i}-\mathcal{i}\circ\mathbf{D}$;
combining these observation we conclude that

$R_{A}\circ\widehat{R}_{A}=\text{id}+\mathcal{i}\circ
S_{-}\circ\mathcal{p}\circ A-S_{-}^{G}\circ
A-S_{-}^{G}\circ\left(\mathbf{D}^{G}\circ\mathcal{i}-\mathcal{i}\circ\mathbf{D}\right)\circ
S_{-}\circ\mathcal{p}\circ A\ .$

Due to the properties of the propagators and of the support of $A$, using the
fact that $M$ is globally hyperbolic and the fact that
$S_{-}^{G}\circ\mathbf{D}^{G}=\mathrm{id}_{\pazocal{E}(S_{G}(M))}$ on smooth
functions with past compact support we then infer that

$R_{A}\circ\widehat{R}_{A}=\text{id}_{\pazocal{E}(S_{G}(M)}+\mathcal{i}\circ
S_{-}\circ\mathcal{p}\circ A-S_{-}^{G}\circ A-\mathcal{i}\circ
S_{-}\circ\mathcal{p}\circ A+S_{-}^{G}\circ\mathcal{i}\circ\mathcal{p}\circ
A=\text{id}_{\pazocal{E}(S_{G}(M))}\ .$

In the same way, using the fact that

$(\mathcal{p}\circ
A)(s)=ie^{j}(p)\cdot_{\Gamma,G}\left[\tilde{\sigma}(p),{\rho_{G}}_{\ast}\Big{(}\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{j}(p))\Big{)}s^{\sharp_{G}}(\tilde{\sigma}(p))\right]=\mathcal{p}\circ\mathbf{D}^{G}-\mathbf{D}\circ\mathcal{p}$

one can show that

$\widehat{R}_{A}\circ R_{A}=\text{id}_{\pazocal{E}(S(M))}\ .$

We shall thus write $\widehat{R}_{A}$ as $R_{A}^{-1}$.

##  4\. The behaviour of the classical Møller map in the case of a $U(1)$
gauge charge

###  4.1. The classical Møller map and Green operators

If the gauge group $G$ is $U(1)$ the entwining maps $\mathcal{i}$ and
$\mathcal{p}$ defined in Section 3.1 and the classical Møller map on field
configurations $R_{A}$, whose explicit expression is given in Theorem 3.4,
enjoy some further properties which we now explore.

First of all, notice that if we consider the hermitian inner products on
$S(M)$ and $S_{G}(M)$, then $\mathcal{i}$ and $\mathcal{p}$ are the adjoint of
one another: indeed,

$\begin{split}\left(s,(\mathcal{i}t)\right)_{G}(p)&=\left(\rho(\pi_{2}\circ\widehat{\sigma}(p))s^{\sharp_{G}}(\widehat{\sigma}(p)),\rho(\pi_{2}\circ\widehat{\sigma}(p))(\mathcal{i}t)^{\sharp_{G}}(\widehat{\sigma}(p))\right)_{V}\\\
&=\left(\Delta_{r,1}^{\mathbb{C}}\left(\pi_{\mathrm{Spin}}\circ\pi_{2}\circ\widehat{\sigma}(p)\right)\rho(g(p))^{-1}s^{\sharp_{G}}\left(\tilde{\sigma}(p)\right),\Delta_{r,1}^{\mathbb{C}}\left(\pi_{\mathrm{Spin}}\circ\pi_{2}\circ\widehat{\sigma}(p)\right)\rho\left(g(p)\right)^{-1}t^{\sharp}({\sigma}(p))\right)_{V}\\\
&=\left(\Delta_{r,1}^{\mathbb{C}}\left(\pi_{2}\circ\sigma(p)\right)s^{\sharp_{G}}\left(\tilde{\sigma}(p)\right),\Delta_{r,1}^{\mathbb{C}}\left(\pi_{2}\circ\sigma(p)\right)t^{\sharp}({\sigma}(p))\right)_{V}\\\
&=\left(\Delta_{r,1}^{\mathbb{C}}\left(\pi_{2}\circ\sigma(p)\right)(\mathcal{p}s)^{\sharp}\left({\sigma}(p)\right),\Delta_{r,1}^{\mathbb{C}}\left(\pi_{2}\circ\sigma(p)\right)t^{\sharp}({\sigma}(p))\right)_{V}\\\
&=\left((\mathcal{p}s),t\right)(p)\end{split}$

where we have used the fact that $G=U(1)$.

We now move on to the behaviour of the map
$A\colon\pazocal{E}(S_{G}(M))\to\pazocal{E}(S_{G}(M))$ defined in (3.6). First
of all, recall that $\mathfrak{u}(n)$ consists of skew-hermitian matrices, and
thus $\mathfrak{u}(1)$ consists of purely imaginary complex numbers;
therefore,

$\begin{split}\left(s,(At)\right)_{G}(p)&=\left(s(p),(At)(p)\right)_{G}\\\
&=\left(s^{\sharp_{G}}(\widehat{\sigma}(p)),i(g^{kj}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{k}(p))\right)t^{\sharp_{G}}(\widehat{\sigma}(p))\right)\right)_{V}\\\
&=-\left(i(g^{kj}(p)e_{j}(p))\cdot_{V}s^{\sharp_{G}}(\widehat{\sigma}(p)),{\rho_{G}}_{\ast}\left(\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{k}(p))\right)t^{\sharp_{G}}(\widehat{\sigma}(p))\right)_{V}\\\
&=\left(i{\rho_{G}}_{\ast}\left(\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{k}(p))\right)\left((g^{kj}(p)e_{j}(p))\cdot_{V}s^{\sharp_{G}}(\widehat{\sigma}(p))\right),t^{\sharp_{G}}(\widehat{\sigma}(p))\right)_{V}\\\
&=\left(i(g^{kj}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(-(\pi_{G}\circ
g)^{\ast}\theta_{G}+(f_{G}\circ\widehat{\sigma})^{\ast}\omega^{G}\right)_{p}(e_{k}(p))\right)s^{\sharp_{G}}(\widehat{\sigma})\right),t^{\sharp_{G}}(\widehat{\sigma})\right)_{V}\\\
&=\left((As)(p),t(p)\right)_{G}=\left((As),t\right)_{G}(p)\ .\end{split}$

Let us then consider the formal adjoint
${R_{A}}^{\ast}\colon\pazocal{E}(\overline{S_{G}(M)})\to\pazocal{E}(\overline{S(M)})$
of the classical Møller map on field configurations, i.e.

$\int_{M}\left((R_{A}t),s\right)_{G}(p)\,d\mu_{g}=\int_{M}\left(t,({R_{A}}^{\ast}s)\right)\,d\mu_{g}\
\text{for every}\ t\in\pazocal{D}(S(M)),s\in\pazocal{D}(\overline{S_{G}(M)})\
.$

Notice that as $R_{A}\colon\pazocal{E}(S(M))\to\pazocal{E}(S_{G}(M))$ is
linear, using the antilinear isomorphisms between a vector bundle and its
conjugate bundle

$C\colon\overline{S(M)}\to{S(M)}\qquad C_{G}\colon\overline{S_{G}(M)}\to
S_{G}(M)$

we can naturally induce a linear map
${\overline{R}_{A}}\colon\pazocal{E}(\overline{S(M)})\to\pazocal{E}(\overline{S_{G}(M)})$,
$\overline{R}_{A}\overset{\cdot}{=}C_{G}^{-1}\circ R_{A}\circ C$, whose formal
adjoint is given by
${\overline{R}_{A}}^{\ast}\colon\pazocal{E}(S_{G}(M))\to\pazocal{E}(S(M))$. In
particular, it holds that ${\overline{R}_{A}}^{\ast}=\overline{R_{A}^{\ast}}$.
In light of the previous equalities, we can write

${\overline{R}_{A}}^{\ast}=\mathcal{p}-\mathcal{p}\circ A\circ S_{+}^{G}\qquad
R_{A}^{\ast}=C^{-1}\circ{\overline{R}_{A}}^{\ast}\circ C_{G}\ .$

Notice that the antilinear isomorphisms can be used to define

$\overline{\mathcal{i}}\colon\pazocal{E}(\overline{S(M)})\to\pazocal{E}(\overline{S_{G}(M)})\qquad\overline{\mathcal{p}}\colon\pazocal{E}(\overline{S_{G}(M)})\to\pazocal{E}(\overline{S(M)})\
.$

These maps, as well as the map
$\overline{R}_{A}\colon\pazocal{E}(\overline{S(M)})\to\pazocal{E}(\overline{S_{G}(M)})$
satisfy the same results as the ones previously proven.

######  Proposition 4.1.

The advanced and retarded propagators of $\mathbf{D}^{G}$ and $\mathbf{D}$ are
related by

$S_{-}^{G}=R_{A}\circ S_{-}\circ\mathcal{p}\qquad S_{+}^{G}=\mathcal{i}\circ
S_{+}\circ{\overline{R}_{A}}^{\ast}\ .$ (4.1)

where the composition of maps appearing on the right-hand sides are restricted
to $\pazocal{D}(S_{G}(M))$.

###### Proof.

Let us denote with $\widehat{S}_{\pm}^{G}$ the operators on the right-hand
sides of the equalities in (4.1). It is easy to see that on
$\pazocal{D}(S_{G}(M))$ it holds that

$\mathbf{D}^{G}\circ\widehat{S}^{G}_{-}=\mathbf{D}^{G}\circ R_{A}\circ
S_{-}\circ\mathcal{p}\overset{\text{Def.}\
\ref{Def:2.3}}{=}\mathcal{i}\circ\mathbf{D}\circ
S_{-}\circ\mathcal{p}=\mathcal{i}\circ\mathcal{p}=\text{id}$

and

$\begin{split}\widehat{S}_{-}^{G}\circ\mathbf{D}^{G}&=R_{A}\circ
S_{-}\circ\mathcal{p}\circ\mathbf{D}^{G}=R_{A}\circ S_{-}\circ\mathbf{D}\circ
R_{A}^{-1}=R_{A}\circ R_{A}^{-1}\\\ &=\text{id}\ .\end{split}$

The same relations for $\widehat{S}_{+}^{G}$ can be obtained in the following
way: we know that ${\mathbf{D}^{G}}^{\ast}=C_{G}^{-1}\circ\mathbf{D}^{G}\circ
C_{G}$ and that
${\widehat{S}_{-}}^{G^{\ast}}=C_{G}^{-1}\circ\widehat{S}_{+}^{G}\circ C_{G}$;
therefore given any $u\in\pazocal{D}(S_{G}(M))$,
$v\in\pazocal{D}(\overline{S_{G}(M)})$

$\begin{split}\int_{M}\left(u,v\right)_{G}\,d\mu_{g}&=\int_{M}\left(\mathbf{D}^{G}\widehat{S}^{G}_{-}u,v\right)_{G}\,d\mu_{g}=\int_{M}\left(\widehat{S}_{-}^{G}u,C_{G}^{-1}\circ\mathbf{D}^{G}\circ
C_{G}v\right)_{G}\,d\mu_{g}\\\
&=\int_{M}\left(u,C_{G}^{-1}\circ\widehat{S}_{+}^{G}\circ\mathbf{D}^{G}\circ
C_{G}v\right)\,d\mu_{g}\ .\end{split}$

As the above equality holds for any functions $u\in\pazocal{D}(S_{G}(M))$,
$v\in\pazocal{D}(\overline{S_{G}(M)})$, we infer that
$\widehat{S}^{G}_{+}\circ\mathbf{D}^{G}=\text{id}_{\pazocal{D}(S_{G}(M))}$.
The other equality can be obtained in a similar fashion.

To conclude the proof, it suffices to show that

${\mathrm{supp}}(\widehat{S}_{\pm}^{G}(u))\subseteq
J^{\mp}({\mathrm{supp}}(u))\ \text{for every}\ u\in\pazocal{D}(S_{G}(M))\ .$

We proceed for the retarded propagator $\widehat{S}_{-}^{G}$, as the proof for
the advanced one is entirely analogous. In particular, we shall prove that

$M\setminus J^{+}({\mathrm{supp}}(u))\subseteq
M\setminus{\mathrm{supp}}(\widehat{S}_{-}^{G}(u))\ .$

To this end, it suffices to show that
$M\setminus{\mathrm{supp}}(S_{-}^{G}(u))\subseteq
M\setminus{\mathrm{supp}}(\widehat{S}_{-}^{G}(u))$; therefore, let
$p\notin{\mathrm{supp}}(S_{-}^{G}(u))$, and let us consider two Cauchy
surfaces $\Sigma_{1},\Sigma_{2}\subseteq M$ such that

  1. $(i)$

$\Sigma_{2}\subseteq J^{+}(\Sigma_{1})$;

  2. $(ii)$

$\Sigma_{1}\cap\Sigma_{2}=\varnothing$;

  3. $(iii)$

$\left(\mathrm{supp}(\mathscr{A})\cup\\{p\\}\cup{\mathrm{supp}}(u)\right)\cap
J^{+}(\Sigma_{1})=\varnothing$.

We then consider a smooth function $\varphi\in\pazocal{E}(M)$ such that

$\varphi\equiv 1\ \text{on}\ J^{-}(\Sigma_{1}),\qquad\varphi\equiv 0\
\text{on}\ J^{+}(\Sigma_{2})$

which we use to define the maps

$\displaystyle\varphi\colon\pazocal{E}(S(M))$
$\displaystyle\to\pazocal{E}(S(M))\qquad\qquad\qquad\qquad\varphi_{G}\colon\pazocal{E}(S_{G}(M))$
$\displaystyle\to\pazocal{E}(S_{G}(M))$ $\displaystyle s\quad$
$\displaystyle\mapsto(\varphi
s)(p)\overset{\cdot}{=}\varphi(p)s(p)\qquad\qquad\qquad\qquad t$
$\displaystyle\mapsto(\varphi_{G}t)(p)\overset{\cdot}{=}\varphi(p)t(p)$

as well as the analogous maps $1-\varphi$ and $1-\varphi_{G}$. Notice that
$\mathcal{i}\circ\varphi=\varphi_{G}\circ\mathcal{i}$ and
$\mathcal{p}\circ\varphi_{G}=\varphi\circ\mathcal{p}$ and that
${\mathrm{supp}}(\varphi\circ S_{-}\circ\mathcal{p}(u))$ is compact. Then

$\begin{split}(\widehat{S}_{-}^{G}u)&=\left(R_{A}\circ
S_{-}\circ\mathcal{p}\right)(u)\\\
&=\left(R_{A}\circ\left(1-\varphi+\varphi\right)\circ
S_{-}\circ\mathcal{p}\right)(u)\\\ &=\left(R_{A}\circ\varphi\circ
S_{-}\circ\mathcal{p}\right)(u)+\left(R_{A}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)\\\ &=(\mathcal{i}-S_{-}^{G}\circ
A\circ\mathcal{i})\circ(\varphi\circ
S_{-}\circ\mathcal{p})(u)+\left(R_{A}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)\\\
&=\left(S_{-}^{G}\circ\mathbf{D}^{G}\circ\mathcal{i}\circ\varphi\circ
S_{-}\circ\mathcal{p}\right)(u)-\left(S_{-}^{G}\circ
A\circ\mathcal{i}\circ\varphi\circ
S_{-}\circ\mathcal{p}\right)(u)+\left(R_{A}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)\\\
&=\left(S_{-}^{G}\circ\mathcal{i}\circ\mathbf{D}\circ\varphi\circ
S_{-}\circ\mathcal{p}\right)(u)+\left(R_{A}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)\\\
&=\left(S_{-}^{G}\circ\mathcal{i}\circ\mathbf{D}(\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)+\left(S_{-}^{G}\circ\mathcal{i}\circ\varphi\circ\mathbf{D}\circ
S_{-}\circ\mathcal{p}\right)(u)+\left(R_{A}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)\\\
&=\left(S_{-}^{G}\circ\mathcal{i}\circ\mathbf{D}(\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)+S_{-}^{G}(u)+\left(R_{A}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)\ .\end{split}$

Now, notice that when evaluated at $p$, the above expression is zero: indeed,
we know that $p\notin{\mathrm{supp}}(S_{-}^{G}(u))$, and due to the properties
of $\varphi\in\pazocal{E}(M)$ we also know that $\mathbf{D}(\varphi)\equiv 0$
on $J^{-}(\Sigma_{1})$ and $J^{+}(\Sigma_{2})$, and therefore
$(\mathcal{i}\circ\mathbf{D}(\varphi)\circ S_{-}\circ\mathcal{p})(u)$ is
supported in $J^{+}(\Sigma_{1})\cap J^{-}(\Sigma_{2})$. But then
$S_{-}^{G}((\mathcal{i}\circ\mathbf{D}(\varphi)\circ S_{-}\circ\mathcal{p})u)$
is supported in the causal future of that set; as $p\in J^{-}(\Sigma_{1})$ we
then have that the first term vanishes at $p$. Let us then consider the last
term: we have

$\left(\mathcal{i}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)=\left((1-\varphi_{G})\circ\mathcal{i}\circ
S_{-}\circ\mathcal{p}\right)(u)$

which is equal to zero when evaluated at $p$, and

$-\left(S_{-}^{G}\circ A\circ\mathcal{i}\circ(1-\varphi)\circ
S_{-}\circ\mathcal{p}\right)(u)=-\left(S_{-}^{G}\circ
A\circ(1-\varphi_{G})\circ\mathcal{i}\circ S_{-}\circ\mathcal{p}\right)(u)$

which vanishes as
$\mathrm{supp}(\mathscr{A})\cap{\mathrm{supp}}(1-\varphi)=\varnothing$. Thus
$p\notin{\mathrm{supp}}(\widehat{S}_{-}^{G}(u))$, and we conclude. ∎

######  Corollary 4.2.

The causal propagators of $\mathbf{D}^{G}$ and $\mathbf{D}$ are related by

$S^{G}=R_{A}\circ
S\circ{\overline{R}_{A}}^{\ast}\Big{|}_{\pazocal{D}(S_{G}(M))}$ (4.2)

###### Proof.

Let us define $o\overset{\cdot}{=}-S_{-}\circ\mathcal{p}\circ A$; then

$\begin{split}R_{A}-\mathcal{i}&=R_{A}-R_{A}\circ
R_{A}^{-1}\circ\mathcal{i}=R_{A}\circ\left(\text{id}-R_{A}^{-1}\circ\mathcal{i}\right)=R_{A}\circ\left(\mathcal{p}\circ\mathcal{i}-\left(\mathcal{p}+S_{-}\circ\mathcal{p}\circ
A\right)\circ\mathcal{i}\right)\\\ &=R_{A}\circ(-S_{-}\circ\mathcal{p}\circ
A\circ\mathcal{i})=R_{A}\circ o\circ\mathcal{i}\ .\end{split}$

Analogously, $R_{A}-\mathcal{i}=\mathcal{i}\circ o\circ R_{A}$; moreover, we
also have the same relations concerning the formal adjoints:

${\overline{R}_{A}}^{\ast}-\mathcal{p}={\overline{R}_{A}}^{\ast}\circ
o^{\dagger}\circ\mathcal{p}\qquad{\overline{R}_{A}}^{\ast}-\mathcal{p}=\mathcal{p}\circ
o^{\dagger}\circ{\overline{R}_{A}}^{\ast}$

where $o^{\dagger}\overset{\cdot}{=}-A\circ\mathcal{i}\circ S_{+}$. Now, using
these equalities we have on $\pazocal{D}(S_{G}(M))$

$R_{A}\circ(S_{-}-S_{+})\circ{\overline{R}_{A}}^{\ast}=\left(R_{A}\circ
S_{-}\circ(\mathcal{p}+\mathcal{p}\circ
o^{\dagger}\circ\overline{R_{A}}^{\ast})-(\mathcal{i}+R_{A}\circ
o\circ\mathcal{i})\circ S_{+}\circ{\overline{R}_{A}}^{\ast}\right)\ .$

Proposition 4.1 then entails that on $\pazocal{D}(S_{G}(M))$ we have that

$\begin{split}R_{A}\circ S\circ{\overline{R}_{A}}^{\ast}&=S_{-}^{G}+R_{A}\circ
S_{-}\circ\mathcal{p}\circ(-A\circ\mathcal{i}\circ
S_{+})\circ{\overline{R}_{A}}^{\ast}-S_{+}^{G}-R_{A}\circ
o\circ\mathcal{i}\circ S_{+}\circ{\overline{R}_{A}}^{\ast}\\\
&=S^{G}+R_{A}\circ o\circ i\circ
S_{+}\circ{\overline{R}_{A}}^{\ast}-R_{A}\circ o\circ i\circ
S_{+}\circ{\overline{R}_{A}}^{\ast}=S^{G}\ .\end{split}$

∎

###  4.2. The classical Møller map and Hadamard bidistributions

Having assessed the properties of the classical Møller map on field
configurations when coupled with the Green operators associated to the free
and uncharged Dirac operators $\mathbf{D}$ and $\mathbf{D}^{G}$, we now
examine the behaviour of Hadamard bidistributions when coupled with the Møller
maps.

First of all, let us recall that the algebras associated to the Dirac field
need to account for both the spinor and cospinor field, that is, sections of
both the vector bundle $S(M)$ and its conjugate bundle $\overline{S(M)}$. In
order to do so, one considers the Whitney sum of the two vector bundles
$S(M)\oplus\overline{S(M)}$ and $S_{G}(M)\oplus\overline{S_{G}(M)}$, which we
shall denote with $S^{\oplus}(M)$ and $S^{\oplus}_{G}(M)$ respectively. A
section $u$ of $S^{\oplus}(M)$ can be then understood as a couple
$(u_{1},u_{2})$ with $u_{1}\in\pazocal{E}(S(M))$ and
$u_{2}\in\pazocal{E}(\overline{S(M)})$; the same holds for sections of
$S^{\oplus}_{G}(M)$.

The hermitian metric on $S(M)$, which can be understood as a bilinear map

$\pazocal{E}(S(M))\times\pazocal{E}(\overline{S(M)})\ni
u,v\mapsto(u,v)\in\pazocal{E}(M)$

can be used to induce a symmetric and bilinear map (denoted with the same
symbol)

$\pazocal{E}(S^{\oplus}(M))\times\pazocal{E}(S^{\oplus}(M))\ni
u,v\mapsto(u,v)\overset{\cdot}{=}(v_{1},u_{2})+(u_{1},v_{2})\in\pazocal{E}(M)\
.$

One can also define an involution on the space of sections
$\pazocal{E}(S^{\oplus}(M))$ by using the conjugation maps
$C\colon\overline{S(M)}\to S(M)$ and $C^{-1}\colon S(M)\to\overline{S(M)}$:

$\pazocal{E}(S^{\oplus}(M))\ni u=(u_{1},u_{2})\mapsto
u^{\ast}\overset{\cdot}{=}(Cu_{2},C^{-1}u_{1})\in\pazocal{E}(S^{\oplus}(M))\
.$

Using the Dirac operator
$\mathbf{D}\colon\pazocal{E}(S(M))\to\pazocal{E}(S(M))$ and its adjoint
$\mathbf{D}^{\ast}=C^{-1}\circ\mathbf{D}\circ C$ as well as the causal
propagators $S\colon\pazocal{D}(S(M))\to\pazocal{E}(S(M))$ and $S^{\ast}$ we
construct the operators

$\mathbf{D}^{\oplus}\overset{\cdot}{=}\mathbf{D}\oplus-\mathbf{D}^{\ast}\qquad
S^{\oplus}\overset{\cdot}{=}S\oplus-S^{\ast}$

Notice that $S^{\oplus}$, which is the causal propagator for
$\mathbf{D}^{\oplus}$, is formally self-adjoint: indeed, given
$u,v\in\pazocal{D}(S^{\oplus}(M))$ we have

$\int_{M}(S^{\oplus}u,v)\,d\mu_{g}=\int_{M}(Su_{1},v_{2})-(v_{1},S^{\ast}u_{2})\,d\mu_{g}=\int_{M}-(u_{1},S^{\ast}v_{2})+(Sv_{1},u_{2})\,d\mu_{g}=\int_{M}(u,S^{\oplus}v)\,d\mu_{g}\
.$

Therefore, the distribution
$S^{\oplus}\in\pazocal{D}^{\prime}(S^{\oplus}(M)\boxtimes S^{\oplus}(M))$
given by the Schwartz kernel theorem,

$S^{\oplus}(u,v)\overset{\cdot}{=}\int_{M}(S^{\oplus}u,v)\,d\mu_{g}$

is symmetric. Analogous extensions can be made in the case of the charged
spinor bundle $S_{G}(M)$.

The Møller map $R_{A}\colon\pazocal{E}(S(M))\to\pazocal{E}(S(M))$ as well as
its conjugate
$\overline{R}_{A}\colon\pazocal{E}(\overline{S(M)})\to\pazocal{E}(\overline{S(M)})$,
which satisfy Definition 3.3 and Theorem 3.4 (with suitable modifications),
can be combined into one Møller map
$\mathcal{R}_{A}\colon\pazocal{E}(S^{\oplus}(M))\to\pazocal{E}(S_{G}^{\oplus}(M))$,

$\mathcal{R}_{A}\overset{\cdot}{=}R_{A}\oplus\overline{R}_{A}=i^{\oplus}-{S_{-}^{G}}^{\oplus}\circ
A^{\oplus}\circ i^{\oplus}\ .$

The same can be done with the formal adjoints of the Møller maps, yielding

${\mathcal{R}_{A}}^{\ast}=\overline{R}_{A}^{\ast}\oplus R_{A}^{\ast}\ .$

We now recall the definition of Hadamard bidistribution for spinor fields.

######  Definition 4.3.

A distribution
$\omega\in\pazocal{D}^{\prime}(S^{\oplus}(M)\boxtimes{S^{\oplus}(M)})$
satisfies the Hadamard two-point condition if given any two sections
$u,v\in\pazocal{D}(S^{\oplus}(M))$ we have:

  1. $(i)$

$\omega((\mathbf{D}^{\oplus}u),{v})=0$;

  2. $(ii)$

$\omega(u,v)+\omega(v,u)=iS^{\oplus}(u,{v})$;

  3. $(iii)$

we require that

$\text{WF}(\omega)=\left\\{(x,y,\xi_{x},-\xi_{y})\in T^{\ast}M^{2}\setminus
z(M^{2})\ |\ (x,\xi_{x})\sim(y,\xi_{y})\ \mathrm{or}\ x=y,\xi_{x}=\xi_{y},\
\xi_{x}\triangleright 0\right\\}$

where $\xi_{x}\triangleright 0$ means that $\xi_{x}$ is future-directed and
lightlike, and $(x,\xi_{x})\sim(y,\xi_{y})$ means that $x$ can be connected to
$y$ by means of a future-directed lightlike geodesic $\gamma$ such that
$\xi_{x}$ is the cotangent vector of $\gamma$ at $x$ and $\xi_{y}$ is the
cotangent vector of $\gamma$ at $y$.

Notice that such distributions do exist; see for instance [42] and [31].

######  Proposition 4.4.

If $\omega\in\pazocal{D}^{\prime}(S^{\oplus}(M)\boxtimes{S^{\oplus}(M)})$ is a
distribution satisfying the Hadamard two-point condition, then

$\omega_{G}(\cdot,\cdot)\overset{\cdot}{=}\omega({\mathcal{R}_{A}}^{\ast}\cdot,{\mathcal{R}_{A}}^{\ast}\cdot)$
(4.3)

is a distribution in
$\pazocal{D}^{\prime}(S_{G}^{\oplus}(M)\boxtimes{S_{G}^{\oplus}(M)})$
satisfying the Hadamard two-point condition.

###### Proof.

First of all, notice that $\omega_{G}$ defined as in (4.3) is well-defined:
indeed, $\overline{R}_{A}^{\ast}$ and ${R_{A}}^{\ast}$ are continuous with
respect to the inductive limit topology of $\pazocal{D}(S_{G}(M))$ and
$\pazocal{D}(\overline{S_{G}(M)})$, being linear maps which are sequentially
continuous. Moreover, they evidently map compactly supported smooth functions
to compactly supported smooth functions. These result directly translate to
the map ${\mathcal{R}_{A}}^{\ast}$. Let us now prove that the three
requirements are satisfied:

  1. $(i)$

it is easy to see that $\omega_{G}({\mathbf{D}^{G}}^{\oplus}u,{v})=0$; indeed,
by Definition 3.3 and Theorem 3.4 we have

$R_{A}^{\ast}\circ{\mathbf{D}^{G}}^{\ast}=(\mathbf{D}^{G}\circ
R_{A})^{\ast}=(\mathcal{i}\circ\mathbf{D})^{\ast}=\mathbf{D}^{\ast}\circ\mathcal{i}^{\ast}$

and therefore
${\mathcal{R}_{A}}^{\ast}\circ{\mathbf{D}^{G}}^{\oplus}=(\mathbf{D}\circ\mathcal{p})\oplus(-\mathbf{D}^{\ast}\circ\mathcal{i}^{\ast})$;
this entails that

$\omega_{G}({\mathbf{D}^{G}}^{\oplus}u,{v})=\omega\left({{\mathcal{R}_{A}}}^{\ast}({\mathbf{D}^{G}}^{\oplus}u),{\mathcal{R}_{A}}^{\ast}{v}\right)=\omega(\mathbf{D}^{\oplus}((\mathcal{p}\oplus\mathcal{i}^{\ast})u),{\mathcal{R}_{A}}^{\ast}{v})=0\
.$

  2. $(ii)$

We know that
$\omega_{G}(u,{v})=\omega({\mathcal{R}_{A}}^{\ast}u,{\mathcal{R}_{A}}^{\ast}{v})$,
and by using Corollary 4.2 we have

$\begin{split}\omega_{G}(u,{v})+\omega_{G}(v,{u})&=\omega({\mathcal{R}_{A}}^{\ast}u,{\mathcal{R}_{A}}^{\ast}{v})+\omega({\mathcal{R}_{A}}^{\ast}v,{\mathcal{R}_{A}}^{\ast}{u})\\\
&=iS^{\oplus}({\mathcal{R}_{A}}^{\ast}u,{\mathcal{R}_{A}}^{\ast}{v})\\\
&=i\int_{M}\left(S^{\oplus}({\mathcal{R}_{A}}^{\ast}u),{\mathcal{R}_{A}}^{\ast}{v}\right)\,d\mu_{g}\\\
&=i\int_{M}\left(S\circ\overline{R}_{A}^{\ast}u_{1},R_{A}^{\ast}v_{2}\right)-\left(\overline{R}_{A}^{\ast}v_{1},S^{\ast}\circ{R}_{A}^{\ast}u_{2}\right)\,d\mu_{g}\\\
&=i\int_{M}\left((R_{A}\circ
S\circ\overline{R}_{A}^{\ast})u_{1},v_{2}\right)-\left(v_{1},(\overline{R}_{A}\circ
S^{\ast}\circ{R}_{A}^{\ast})u_{2}\right)\,d\mu_{g}\\\
&=i\int_{M}\left({S^{G}}^{\oplus}u,{v}\right)_{G}\,d\mu_{g}=i{S^{G}}^{\oplus}(u,{v})\
.\end{split}$

  3. $(iii)$

We know that $\omega_{G}$ is a bisolution of ${\mathbf{D}^{G}}^{\oplus}$, and
that the principal symbol of ${\mathbf{D}^{G}}^{\oplus}$ coincides with that
of $\mathbf{D}^{\oplus}$; therefore by [15, Theorem 6.1.1] we know that the
wavefront set of $\omega_{G}$ coincides with that of $\omega$, being
determined by the Hamiltonian flow associated to the principal symbol of
${\mathbf{D}^{G}}^{\oplus}$.

∎

##  5\. The classical Møller map on the observable algebras

###  5.1. The Poisson $\ast$-algebras of observables

Our goal is to pass the classical Møller map on field configurations $R_{A}$,
presented in Definition 3.3 and Theorem 3.4, to the algebras of observables of
the charged and uncharged Dirac field. We briefly recall the construction of
said topological algebras, as presented in [35], [42] and in Section 2 of [8].
Let $E\overset{\pi}{\to}M$ be a vector bundle with typical fiber $V$, be it
either $S(M)\oplus\overline{S(M)}$ or $S_{G}(M)\oplus\overline{S_{G}(M)}$,
endowed with a symmetric bilinear metric $h$, and let us consider the exterior
algebra of $\pazocal{E}(E)$, that is, the graded algebra

$\wedge^{\bullet}\pazocal{E}(E)=\bigoplus_{p\in\mathbb{N}}\wedge^{p}\pazocal{E}(E)\
.$

We can consider the spaces of homogeneous elements $\wedge^{p}\pazocal{E}(E)$
as embedded into $\Gamma(M^{p},E^{\boxtimes
p})\simeq\overline{\Gamma(M,E)^{{\otimes}p}}$; then using the usual Fréchet
topology (uniform convergence of all derivatives on compact sets) on
$\Gamma(M^{p},E^{\boxtimes p})$ we define the spaces of $p$-antisymmetric
sections

$\pazocal{E}^{a}(M^{p},E^{\boxtimes
p})\overset{\cdot}{=}\overline{\wedge^{p}\pazocal{E}(E)}$

as well as the configuration space

$\mathcal{C}(E)\overset{\cdot}{=}{\widehat{\bigoplus}_{p\in\mathbb{N}}\pazocal{E}^{a}(M^{p},E^{\boxtimes
p})}$

where $\widehat{\bigoplus}$ denotes the algebraic direct sum. Notice that the
involution defined in Section 4.2 can be extended to an involution
${\cdot}^{\ast}\colon\mathcal{C}(E)\to\mathcal{C}(E)$ by requiring the
behaviour

$(u_{1}\wedge\cdots\wedge
u_{p})^{\ast}\overset{\cdot}{=}u_{p}^{\ast}\wedge\cdots\wedge u_{1}^{\ast}$

on homogeneous elements in $\wedge^{p}\pazocal{E}(E)$, by extending the above
by continuity to $\pazocal{E}^{a}(M^{p},E^{\boxtimes p})$ and by linearity to
$\mathcal{C}(E)$.

We are interested in _antisymmetric functionals_ on the space of sections
$\pazocal{E}(E)$; these can be interpreted as a sequence
$\\{F_{p}\\}_{p\in\mathbb{N}}$ of linear and continuous functionals on
$\\{\wedge^{p}\pazocal{E}(E)\\}_{p\in\mathbb{N}}$, that is, a sequence of
elements such that

$F_{p}\in\mathcal{F}^{p}(E)\overset{\cdot}{=}{\pazocal{E}^{a}}^{\prime}(M^{p},E^{\boxtimes
p})\ \ \text{for all}\ p\in\mathbb{N}\ ,$

where the $(\cdot)^{\prime}$ means the strong topological dual.

Thus we define the space of _fermionic functionals_ as

$\mathcal{F}(E)\overset{\cdot}{=}\prod_{p\in\mathbb{N}}\mathcal{F}^{p}(E)\ .$

There exists a duality pairing between $\mathcal{F}(E)$ and $\mathcal{C}(E)$
given by

$\langle F,u\rangle\overset{\cdot}{=}\sum_{p\in\mathbb{N}}\langle
F_{p},u_{p}\rangle\ \text{for all}\ F\in\mathcal{F}(E),\ u\in\mathcal{C}(E)\
.$

Notice that the sum is finite and thus always well-defined, $\mathcal{C}(E)$
being an algebraic direct sum. We can endow $\mathcal{F}(E)$ with the weak
topology $\tau_{\sigma}$, that is, the topology given by the family of
seminorms $\\{p_{u}\\}_{u\in\mathcal{C}(E)},\ p_{u}(F)=\left\lvert
F(u)\right\rvert$, thus making it a locally convex topological vector space
which happens to be nuclear and sequentially complete.
$\mathcal{F}(E)$ can be endowed with an antisymmetric, pointwise product
initially defined on homogeneous elements in $\wedge^{p}\pazocal{E}(E)$ by

$(F\wedge G)_{p}(u_{1}\wedge\cdots\wedge
u_{p})\overset{\cdot}{=}\sum_{\sigma\in
S_{p}}\mathrm{sgn}(\sigma)\sum_{k=0}^{p}\frac{1}{k!(p-k)!}F_{k}(u_{\sigma(1)}\wedge\cdots\wedge
u_{\sigma(k)})G_{p-k}(u_{\sigma(k+1)}\wedge\cdots\wedge u_{\sigma(p)})\ .$

The object above is then extended by linearity and continuity to elements in
$\pazocal{E}^{a}(M^{p},E^{\boxtimes p})$, thus yielding a well-defined object
in $\mathcal{F}(E)$. Moreover, said product is continuous with respect to the
topology on $\mathcal{F}(E)$. $\mathcal{F}(E)$ is also naturally endowed with
an involution ${\cdot}^{\ast}\colon\mathcal{F}(E)\to\mathcal{F}(E)$,

$\\{F_{p}\\}_{p\in\mathbb{N}}\mapsto\left\\{F_{p}^{\ast}\right\\}_{p\in\mathbb{N}},\qquad(F_{p}^{\ast})(u_{p})\overset{\cdot}{=}\overline{F_{p}(u_{p}^{\ast})}\
.$ (5.1)

One can then consider derivatives of fermionic functionals in the following
way: given $F_{p}\in\mathcal{F}^{p}(E)$, $p\geq 1$, the left derivative of
$F_{p}$ in the direction $h\in\pazocal{E}(E)$ is defined on
$\wedge^{p-1}\pazocal{E}(E)$ as

$d_{h}F_{p}(u)\overset{\cdot}{=}F_{p}(h\wedge u)$

and is then extended by continuity to $\pazocal{E}^{a}(M^{p-1},E^{\boxtimes
p-1})$, thus yielding a linear and continuous map
$d_{h}F_{p}\colon\pazocal{E}^{a}(M^{p-1},E^{\boxtimes p-1})\to\mathbb{C}$,
i.e. $d_{h}F_{p}\in\mathcal{F}^{p-1}(E)$. One can then extend the map
$d_{h}\colon\mathcal{F}^{p}(E)\to\mathcal{F}^{p-1}(E)$ to the whole algebra
$\mathcal{F}(E)$ by considering

$d_{h}\colon F=\\{F_{p}\\}_{p\in\mathbb{N}}\mapsto
d_{h}F\overset{\cdot}{=}\\{d_{h}F_{p}\\}_{p\in\mathbb{N}}\ .$

It is easy to see that for any $h\in\pazocal{E}(E)$, $d_{h}$ is a graded
derivation. One can also consider higher order derivatives by iterating the
left derivative: given $F_{p}\in\mathcal{F}^{p}(E)$, $k\leq p$ and
$h_{1},\dots,h_{k}\in\pazocal{E}(E)$, we define for
$u\in\wedge^{p-k}\pazocal{E}(E)$

$d^{k}_{h_{1},\dots,h_{k}}F_{p}(u)=F_{p}(h_{k}\wedge\cdots\wedge h_{1}\wedge
u)$

and then proceed by continuity as before, obtaining for each
$F\in\mathcal{F}(E)$ a jointly continuous map

$d^{k}F\colon\pazocal{E}(E)^{k}\times\mathcal{C}(E)\to\mathbb{C}$

which is easily seen to be multilinear and alternating in the first $k$
entries, that is, equivalently, a continuous map

$F^{(k)}\colon\pazocal{E}^{a}(M^{k},E^{\boxtimes
k})\times\mathcal{C}(E)\to\mathbb{C}$

which is linear in the first entry. In particular, notice that these can be
considered as an $\mathcal{F}(E)$-valued (compactly supported) distributional
section of ${E^{\ast}}^{\boxtimes k}\to M^{k}$, that is, an object of
$\pazocal{D}^{\prime}(M^{k},E^{\boxtimes
k})\widehat{\otimes}_{\pi}\mathcal{F}(E)$, where $\widehat{\otimes}_{\pi}$
denotes the completion of the tensor product in the projective topology222As
both spaces are nuclear, the completion is independent on the chosen topology;
we choose the projective topology just to fix one..

In order to proceed with the quantization, one needs to restrict the
$\ast$-algebra of fermionic functionals in order to endow it with a suitable
$\star$-product. To do so, one needs to be able to control the wavefront set
of derivatives of the relevant functionals; in particular, we consider the set
of _microcausal_ fermionic functionals $\mathcal{A}(E)\subseteq\mathcal{F}(E)$
consisting of those functionals $F\in\mathcal{F}(E)$ such that

$\text{WF}(F^{(n)}_{u})\subseteq\Xi_{n}\overset{\cdot}{=}T^{\ast}\overset{\cdot}{M^{n}}\setminus\bigcup_{(p_{1},\dots,p_{n})\in
M^{n}}\left(\overline{V_{p_{1}}^{+}}\times\cdots\times\overline{V_{p_{n}}^{+}}\right)\cup\left(\overline{V_{p_{1}}^{-}}\times\cdots\times\overline{V_{p_{n}}^{-}}\right)\
\mathrm{for\ every}\ u\in\mathcal{C}(E)\ .$

We endow this space with the initial locally convex topology induced by the
family of linear maps $\\{\ell_{k,u}\\}$,

$F\overset{\ell_{k,u}}{\longmapsto}\begin{dcases}\langle
F,u\rangle\in\mathbb{C}\,&k=0\\\
F^{(k)}_{u}\in{\pazocal{E}_{\Xi_{k}}^{a}}^{\prime}(M^{k},E^{\boxtimes
k})\,&k\geq 1\end{dcases}$

indexed by an integer $k\in\mathbb{N}$ and a function in the configuration
space $u\in\mathcal{C}(E)$, and where
${\pazocal{E}_{\Xi_{k}}^{a}}^{\prime}(M^{k},E^{\boxtimes k})$ denotes the
inductive limit

$\varinjlim{\pazocal{E}_{\Gamma_{k,n}}^{a}}^{\prime}(M^{k},E^{\boxtimes k})$

with $\\{\Gamma_{k,n}\\}_{n\in\mathbb{N}}$ a sequence of closed cones in
$T^{\ast}M^{k}$ such that $\Gamma_{k,n}\subset\overset{\circ}{\Gamma}_{k,n+1}$
and $\cup_{n}\Gamma_{k,n}=\Xi_{k}$, and where
${\pazocal{E}_{\Gamma_{k,n}}^{a}}^{\prime}(M^{k},E^{\boxtimes k})$ is endowed
with the usual Hörmander topology. Using the causal propagator
$S\in\pazocal{D}^{\prime}(E^{\boxtimes 2},M^{2})$ one is then able to endow
$\mathcal{A}(E)$ with a Peierls’ bracket. First of all, given any homogeneous
functional $F\in\mathcal{A}^{p}(E)$, we define the object $(S\ast F^{(1)})$,

$(S\ast
F^{(1)})(u)\overset{\cdot}{=}\int_{M}({S(x,y),F^{(1)}_{u}(y)})_{h}\,d\mu_{g}(y)$
(5.2)

which is well-defined thanks to the wavefront set properties of both the
causal propagator $S$ and of $F^{(1)}_{u}$, as

$\mathrm{WF}(S)=\left\\{(x,y,\xi_{x},-\xi_{y})\in T^{\ast}M^{2}\setminus
z(M^{2})\ |\ (x,\xi_{x})\sim(y,\xi_{y})\right\\}\ .$

Moreover, notice that this object is actually a smooth function: indeed, using
[27, Theorem 8.2.13] one obtains that

$\mathrm{WF}((S\ast
F^{(1)}_{u}))\subseteq\mathrm{WF}_{M}(S)\cup\mathrm{WF}^{\prime}(S)\circ\mathrm{WF}(F^{(1)}_{u})=\varnothing\
\mathrm{for\ all}\ u\in\pazocal{E}^{a}(M^{p},E^{\boxtimes p})\ .$

Therefore, we can consider it as an object in $\pazocal{E}(E^{\ast})\,{\simeq\
}\pazocal{E}(E)$, where the isomorphism is due to the existence of the
symmetric bilinear metric $h$. We can then compute, for any other homogeneous
functional $G\in\mathcal{A}^{q}(E)$, the object

$G^{(1)}\wedge(S\ast F^{(1)})\in\mathcal{A}^{p+q-2}(E)$

which is defined on an element $u_{1}\wedge\cdots\wedge u_{p+q-2}$

$\begin{split}G^{(1)}&\wedge(S\ast F^{(1)})(u_{1}\wedge\cdots\wedge
u_{p+q-2})\\\ &\overset{\cdot}{=}(-1)^{q+1}\sum_{\sigma\in
S_{p+q-2}}\mathrm{sgn}(\sigma)G\left((S\ast
F^{(1)}_{u_{\sigma(q)}\wedge\cdots\wedge u_{\sigma(q+p-2)}})\wedge
u_{\sigma(1)}\wedge\cdots\wedge u_{\sigma(q-1)}\right)\end{split}$

and then extended to the whole of $\pazocal{E}^{a}(M^{p+q-2},E^{\boxtimes
p+q-2})$ by continuity. This procedure is then extended to non-homogeneous
functionals by considering $G^{(1)}\wedge(S\ast
F^{(1)})=\left\\{\left(G^{(1)}\wedge(S\ast
F^{(1)})\right)_{p}\right\\}_{p\in\mathbb{N}}$,

$\left(G^{(1)}\wedge(S\ast
F^{(1)})\right)_{p}(u)\overset{\cdot}{=}\sum_{k=0}^{p}\frac{1}{k!(p-k)!}\left((G^{(1)})_{k}\wedge\left(S\ast(F^{(1)})_{p-k}\right)\right)(u),\qquad
u\in\pazocal{E}^{a}(M^{p},E^{\boxtimes p})\ .$ (5.3)

Then the Peierls’ bracket is given by
$\\{F,G\\}_{S}\overset{\cdot}{=}G^{(1)}\wedge(S\ast F^{(1)})$; notice that on
homogeneous elements $F\in\mathcal{A}^{p}(E)$, $G\in\mathcal{A}^{q}(E)$ we
have the desired graded anticommutativity

$\left\\{G,F\right\\}_{S}=-(-1)^{qp}\left\\{F,G\right\\}_{S}$

as well as the graded Jacobi identity.

###  5.2. Deformation quantization and the classical Møller maps

Starting from $\mathcal{A}(E)$, the quantization proceeds in the following
way: we consider the $\ast$-algebra $\mathcal{A}(E)[\\![\hbar]\\!]$ of formal
power series in $\hbar$ with coefficients in $\mathcal{A}(E)$, endowed with
the product topology; clearly
$\mathcal{A}(E)\subseteq\mathcal{A}(E)[\\![\hbar]\\!]$. One is then able to
introduce a $\star$-product on $\mathcal{A}(E)[\\![\hbar]\\!]$, that is, a
product such that given $F\in\mathcal{A}^{p}(E)$, $G\in\mathcal{A}^{q}(E)$

$F\star G=F\wedge G+o(\hbar)\qquad F\star G-(-1)^{pq}G\star
F=i\hbar\left\\{F,G\right\\}_{S}+o(\hbar^{2})$

Given a Hadamard bidistribution
$\omega\in\pazocal{D}^{\prime}(M^{2},E^{\boxtimes 2})$ (see Definition 4.3)
and two functionals $F,G\in\mathcal{A}(E)$, we consider the fermionic
functional

$\begin{split}\Gamma_{\omega}^{n}(G,F)&\overset{\cdot}{=}\left(\frac{i}{2}\right)^{n}G^{(n)}\wedge\left(\omega^{\boxtimes
n}\ast F^{(n)}\right)\ \forall\ n\in\mathbb{N}\ .\end{split}$

where the quantity on the right is defined component-wise as in (5.3) and
$\omega^{\boxtimes n}\ast F^{(n)}$ is defined as in (5.2). Notice that due to
the wavefront set properties of both $F^{(n)}$ and $\omega^{\boxtimes n}$ the
object above is well-defined. Then we define

$G\star
F\overset{\cdot}{=}\sum_{n\in\mathbb{N}}\hbar^{n}\Gamma_{\omega}^{n}(G,F)\in\mathcal{A}(E)[\\![\hbar]\\!]\
.$ (5.4)

This product can then be easily extended to the whole topological
$\ast$-algebra of formal power series. Now, let
$\mathcal{A}(S)[\\![\hbar]\\!]$ and $\mathcal{A}(S_{G})[\\![\hbar]\\!]$ be the
topological $\ast$-algebras of microcausal fermionic functionals associated to
the free and charged Dirac fields. Suppose that the $\star_{G}$-product on
$\mathcal{A}(S_{G})[\\![\hbar]\\!]$ is constructed using the Hadamard
bidistribution induced by the one used to define $\star$-product on
$\mathcal{A}(S)[\\![\hbar]\\!]$, as illustrated in Proposition 4.4. We define
the _classical Møller map_
$\mathscr{R}_{A}\colon\mathcal{A}(S_{G})[\\![\hbar]\\!]\to\mathcal{A}(S)[\\![\hbar]\\!]$
by considering the pullbacks induced by $\mathcal{R}_{A}^{\wedge
p}\colon\pazocal{E}^{a}(M^{p},S^{\oplus}(M)^{\boxtimes
p})\to\pazocal{E}^{a}(M^{p},S_{G}^{\oplus}(M)^{\boxtimes p})$. Now,

######  Theorem 5.1.

$\mathscr{R}_{A}$ is a well-defined $\ast$-isomorphism, algebraically and
topologically.

###### Proof.

First of all, we need to show that given a functional
$F\in\mathcal{A}(S_{G})\subseteq\mathcal{A}(S_{G})[\\![\hbar]\\!]$,
$\mathscr{R}_{A}(F)$ is a well-defined functional in
$\mathcal{A}(S)[\\![\hbar]\\!]$; that is, we need to show that

$\text{WF}\left((\mathscr{R}_{A}(F))_{u}^{(n)}\right)\subseteq\Xi_{n}\
\mathrm{for\ all}\ u\in\mathcal{C}(S^{\oplus}(M))\ \mathrm{and}\
n\in\mathbb{N}.$ (5.5)

To do so, we need to compute the wavefront set of the classical Møller map on
field configurations $\mathcal{R}_{A}$, restricted to a continuous map
$\mathcal{R}_{A}\colon\pazocal{D}(S^{\oplus}(M))\to\pazocal{D}^{\prime}({S_{G}^{\oplus}(M)})$.
This is defined as the wavefront set of the distribution $r_{A}$, where
$r_{A}\in\pazocal{D}^{\prime}(S^{\oplus}(M)\boxtimes{S_{G}^{\oplus}(M)})$ is
obtained thanks to Schwartz’s kernel theorem and satisfies

$\int_{M}\left(\mathcal{R}_{A}u,v\right)_{G}\,d\mu_{g}=r_{A}(u,v)\ \text{for
every}\ u\in\pazocal{D}(S^{\oplus}(M)),v\in\pazocal{D}({S_{G}^{\oplus}(M)})\
.$

To compute its wavefront set, we proceed locally, as presented in [27].
Namely, let $\\{e_{i}\\}_{1\leq i\leq\text{rank}(S_{G}(M))}$ be a
local333Notice that as we are supposing that the principal bundle $P_{G}$ is
trivial, there exist _global_ frames for both $S(M)$ and $S_{G}(M)$. frame for
$S_{G}(M)$ and $\\{f_{j}\\}_{1\leq j\leq\text{rank}(S(M))}$ a local frame for
$S(M)$; a local frame for $S^{\oplus}(M)$ and $S_{G}^{\oplus}(M)$ is then
given by

$\\{f_{1},\dots,f_{\mathrm{rank}(S(M))},\overline{f}_{1},\dots,\overline{f}_{\mathrm{rank}(S(M))}\\}\qquad\\{e_{1},\dots,e_{\mathrm{rank}(S_{G}(M))},\overline{e}_{1},\dots,\overline{e}_{\mathrm{rank}(S_{G}(M))}\\}$
(5.6)

respectively. Using these, we can write locally

$r_{A}={r_{A}}_{ji}{f}^{j}\boxtimes e^{i}$

with ${r_{A}}_{ij}\in\pazocal{D}^{\prime}(M^{2})$ and where $\\{e^{i}\\}_{i}$
and $\\{f^{j}\\}_{j}$ denote the dual frames of
$S_{G}^{\oplus}(M)^{\ast}\simeq\overline{S_{G}(M)}\oplus S_{G}(M)$ and
$S^{\oplus}(M)^{\ast}\simeq\overline{S(M)}\oplus S(M)$ with respect to those
in (5.6). Then

$\text{WF}(\mathcal{R}_{A})\doteq\text{WF}(r_{A})=\bigcup_{\begin{subarray}{c}1\leq
i\leq 2\text{rank}(S_{G}(M))\\\ 1\leq j\leq
2\text{rank}(S(M))\end{subarray}}\text{WF}\left({r_{A}}_{ji}\right)$

where $\text{WF}({r_{A}}_{ji})$ is given locally by

${\widehat{\phi}_{\alpha}}^{-1}(\text{WF}(\phi_{\alpha}^{\ast}{r_{A}}_{ji}))$

with $\widehat{\phi}_{\alpha}\colon T^{\ast}M_{U_{\alpha}}^{2}\to
U_{\alpha}^{2}\times\mathbb{R}^{\text{dim}(M^{2})}$. Moreover, due to the
properties of the wavefront set, we can study $\text{WF}(\mathcal{R}_{A})$ by
studying separately the two terms

$\mathcal{i}^{\oplus}\ \mathrm{and}\ S_{-}^{\oplus}\circ
A^{\oplus}\circ\mathcal{i}^{\oplus}\ .$

Let us thus first consider the map
$\mathcal{i}^{\oplus}\colon\pazocal{D}(S^{\oplus}(M))\to\pazocal{D}^{\prime}({S_{G}^{\oplus}(M)})$;
then we have that

$\begin{split}\int_{M}\left(\mathcal{i}^{\oplus}u,v\right)_{G}\,d\mu_{g}&=\int_{M}\left(iu_{1},v_{2}\right)_{G}+\left(v_{1},\overline{i}u_{2}\right)_{G}\,d\mu_{g}\\\
&=\int_{M}\left(u_{1}^{\sharp}(\sigma(p)),v_{2}^{\sharp_{G}}(\tilde{\sigma}(p))\right)_{V}\,d\mu_{g}+\int_{M}\left(v_{1}^{\sharp_{G}}(\tilde{\sigma}(p)),u_{2}^{\sharp}(\sigma(p))\right)_{V}\,d\mu_{g}\\\
&=\int_{M}{u_{1}^{\sharp}}^{j}(\sigma(p))h_{ji}{v_{2}^{\sharp_{G}}}^{i}(\tilde{\sigma}(p))\,d\mu_{g}+\int_{M}{v_{1}^{\sharp_{G}}}^{i}(\tilde{\sigma}(p))h_{ij}{u_{2}^{\sharp}}^{j}(\sigma(p))\,d\mu_{g}\end{split}$

that is,

$i_{ji}=\begin{dcases}\delta_{\Delta}h_{ji}\,&j\leq\mathrm{rank}(S(M)),i\leq\mathrm{rank}(S_{G}(M))\\\
\delta_{\Delta}\overline{h}_{(j-\mathrm{rank}(S(M)))(i-\mathrm{rank}(S_{G}(M))}\,&j>\mathrm{rank}(S(M)),i>\mathrm{rank}(S_{G}(M))\end{dcases}\
.$

Therefore,

$\text{WF}(\mathcal{i}^{\oplus})=\left\\{(x,x,k,-k)\in T^{\ast}M^{2}\setminus
z(M^{2})\right\\}\ .$

As far as the map $S_{-}^{\oplus}\circ A^{\oplus}\circ\mathcal{i}^{\oplus}$ is
concerned, we just need to compute the wavefront set of $A^{\oplus}$. In
particular, proceeding as before and using our preferred gauge
$\tilde{\sigma}$ we have

$\begin{split}&\int_{M}(Au,v)_{G}\,d\mu_{g}=\int_{M}\left(i(g^{ij}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)u_{1}^{\sharp_{G}}(\tilde{\sigma}(p))\right),v_{2}^{\sharp_{G}}(\tilde{\sigma}(p))\right)_{V}\,d\mu_{g}\\\
&+\int_{M}\left(v_{1}^{\sharp_{G}}(\tilde{\sigma}(p)),\overline{i(g^{ij}(p)e_{j}(p))}\cdot_{V}\left(\overline{{\rho_{G}}_{\ast}\left(\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)}u_{2}^{\sharp_{G}}(\tilde{\sigma}(p))\right)\right)_{V}\,d\mu_{g}\\\
&=\int_{M}\left(i(g^{ij}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)\right)\right)^{l}_{j}{u_{1}^{\sharp}}^{j}(\tilde{\sigma}(p))h_{li}{v_{2}^{\sharp_{G}}}^{i}(\tilde{\sigma}(p))\,d\mu_{g}\\\
&\quad+\int_{M}\left(\overline{i(g^{ij}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)\right)}\right)^{l}_{i}{v_{1}^{\sharp_{G}}}^{i}(\tilde{\sigma}(p)){u_{2}^{\sharp_{G}}}^{j}(\tilde{\sigma}(p))h_{lj}\
.\end{split}$

Thus

$A_{ji}=\begin{dcases}\delta_{\Delta}\left(i(g^{ij}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)\right)\right)^{l}_{j}h_{li}\,&j,i\leq\mathrm{rank}(S_{G}(M))\\\
\delta_{\Delta}\left(\overline{i(g^{ij}(p)e_{j}(p))\cdot_{V}\left({\rho_{G}}_{\ast}\left(\left(\sigma_{G}^{\ast}\omega^{G}\right)_{p}(e_{i}(p))\right)\right)}\right)^{l}_{i-\mathrm{rank}(S_{G}(M))}\overline{h}_{(j-\mathrm{rank}(S(M))l}\,&j,i>\mathrm{rank}(S_{G}(M))\
.\end{dcases}$

Using the fact that $\sigma_{G}^{\ast}\omega^{G}$ is assumed to be compactly
supported in $\mathrm{supp}(\mathscr{A})$, we then have that

$\begin{split}\text{WF}(A^{\oplus})&\subseteq\left(\pi^{-1}_{T^{\ast}M^{2}}(\mathrm{supp}(\mathscr{A})\times
M)\setminus z(M^{2})\right)\cap\left\\{(x,x,k,-k)\in T^{\ast}M^{2}\setminus
z(M^{2})\right\\}\\\ &=\left\\{(x,x,k,-k)\in T^{\ast}M^{2}\setminus z(M^{2})\
|\ x\in\mathrm{supp}(\mathscr{A})\right\\}\ .\end{split}$

We are now in a position to compute $\text{WF}(\mathcal{R}_{A})$. First of
all, notice that the composition $A^{\oplus}\circ\mathcal{i^{\oplus}}$ is, as
we expected, well-defined: indeed, as
${\mathrm{supp}}(\mathcal{i}^{\oplus})\subseteq\Delta$, we have that
${\mathrm{supp}}(\mathcal{i}^{\oplus})\ni(x,y)\mapsto y$ is proper, and
moreover

$\begin{split}\text{WF}^{\prime}(\mathcal{i}^{\oplus})_{M}&=\left\\{(y,\xi_{y})\
|\ (x,y,0,-\xi_{y})\in\text{WF}(\mathcal{i}^{\oplus})\right\\}=\varnothing\\\
\text{WF}(A^{\oplus})_{M}&=\left\\{(x,\xi_{x})\ |\
(x,y,\xi_{x},0)\in\text{WF}(A^{\oplus})\right\\}=\varnothing\ .\end{split}$

Thus
$\text{WF}^{\prime}(\mathcal{i}^{\oplus})_{M}\cap\text{WF}(A^{\oplus})_{M}=\varnothing$,
and [27, Theorem 8.2.14] gives us the well-posedness of
$A^{\oplus}\circ\mathcal{i}^{\oplus}$ as well as

$\text{WF}(A^{\oplus}\circ\mathcal{i}^{\oplus})\subseteq\left\\{(x,x,k,-k)\in
T^{\ast}M^{2}\setminus z(M^{2})\ |\ x\in\mathrm{supp}(\mathscr{A})\right\\}\
.$

Let us now consider ${S_{-}^{G}}^{\oplus}\circ
A^{\oplus}\circ\mathcal{i}^{\oplus}$; thanks to [34] we know that

$\text{WF}({S_{-}^{G}}^{\oplus})=\left\\{(x,y,\xi_{x},-\xi_{y})\in
T^{\ast}M^{2}\setminus z(M^{2})\ |\ x\in J^{+}(y),(x,\xi_{x})\sim(y,\xi_{y})\
\mathrm{or}\ x=y,\xi_{x}=\xi_{y}\right\\}\ .$

As evidently
$\text{WF}^{\prime}(A^{\oplus}\circ\mathcal{i}^{\oplus})_{M}=\varnothing$ and
${\mathrm{supp}}(A^{\oplus}\circ\mathcal{i}^{\oplus})\ni(x,y)\mapsto y$ is
proper, we can then apply again [27, Theorem 8.2.14] and we have that
$\text{WF}^{\prime}({S_{-}^{G}}^{\oplus}\circ
A^{\oplus}\circ\mathcal{i}^{\oplus})$ is contained in

$\text{WF}^{\prime}({S_{-}^{G}}^{\oplus})\circ\text{WF}^{\prime}(A^{\oplus}\circ\mathcal{i}^{\oplus})\cup\left(\text{WF}({S_{-}^{G}}^{\oplus})_{M}\times
M\times\\{0\\}\right)\cup\left(M\times\\{0\\}\times\text{WF}^{\prime}(A^{\oplus}\circ
i^{\oplus})_{M}\right)\ .$

The first set is given by

$\left\\{(x,z,\xi_{x},\xi_{z})\ |\ \exists(y,\xi_{y})\ \text{s.t.}\
(x,y,\xi_{x},-\xi_{y})\in\text{WF}({S_{-}^{G}}^{\oplus}),\
(y,z,\xi_{y},-\xi_{z})\in\text{WF}(A^{\oplus}\circ\mathcal{i}^{\oplus})\right\\}\
.$

Using the properties of $\text{WF}(A^{\oplus}\circ\mathcal{i}^{\oplus})$ we
conclude that the previous set is given by

$\left\\{(x,z,\xi_{x},\xi_{z})\ |\
(x,z,\xi_{x},-\xi_{z})\in\text{WF}(S_{-}^{G}),\
z\in\mathrm{supp}(\mathscr{A})\right\\}\ .$

The second one and the third one are easily seen to be empty; therefore

$\text{WF}({S_{-}^{G}}^{\oplus}\circ
A^{\oplus}\circ\mathcal{i}^{\oplus})\subseteq\left\\{(x,z,\xi_{x},-\xi_{z})\
|\ (x,z,\xi_{x},-\xi_{z})\in\text{WF}({S_{-}^{G}}^{\oplus}),\
z\in\mathrm{supp}(\mathscr{A})\right\\}\ .$

Therefore

$\text{WF}(\mathcal{R}_{A})\subseteq\left\\{(x,z,\xi_{x},-\xi_{z})\ |\
(x,z,\xi_{x},-\xi_{z})\in\text{WF}({S_{-}^{G}}^{\oplus}),\
z\in\mathrm{supp}(\mathscr{A})\right\\}\cup\left\\{(x,x,k,-k)\in
T^{\ast}M^{2}\setminus z(M^{2})\right\\}\ .$

As the map $(\mathscr{R}_{A}(F))_{u}^{(n)}$ involves the composition of
$F^{(n)}_{\mathcal{R}_{A}u}$ with the map ${\mathcal{R}_{A}}^{\wedge n}$, the
last step consists in computing $\text{WF}(\mathcal{R}_{A}^{\wedge n})$. This,
according to [27, Theorem 8.2.9], is a subset of

$\begin{split}\Big{\\{}&(x_{1},y_{1},x_{2},y_{2},\dots,x_{n},y_{n},\xi_{1},\eta_{1},\xi_{2},\eta_{2},\dots,\xi_{n},\eta_{n})\
|\ \exists I\subseteq\\{1,\dots,n\\},I\neq\varnothing\ \text{s.t.}\\\
&(x_{i},y_{i},\xi_{i},\eta_{i})\in\text{WF}(\mathcal{R}_{A})\ \forall i\in I\
\text{and}\ (x_{j},y_{j},\xi_{j},\eta_{j})=(x_{j},y_{j},0,0)\\\ &\text{with}\
(x_{j},y_{j})\in{\mathrm{supp}}(r_{A})\ \forall j\in\\{1,\dots,n\\}\setminus
I\Big{\\}}\ .\end{split}$

Finally, we are able to check whether $\mathscr{R}_{A}(F)$ is well-defined.
First of all, notice that
$(\mathscr{R}_{A}(F))_{u}^{(n)}=F_{\mathcal{R}_{A}u}^{(n)}\circ{\mathcal{R}_{A}}^{\wedge
n}$ is well-defined: indeed, thanks to [27, Theorem 8.2.13] we know that the
composition is well-defined if
$\text{WF}(F^{(n)}_{\mathcal{R}_{A}u})\cap\text{WF}^{\prime}({\mathcal{R}_{A}}^{\wedge
n})_{M^{n}}=\varnothing$; but

$\left\\{(x_{1},\dots,x_{n},\xi_{1},\dots,\xi_{n})\ |\
(x_{1},y_{1},\dots,x_{n},y_{n},-\xi_{1},0,\dots,-\xi_{n},0)\in\text{WF}({\mathcal{R}_{A}}^{\wedge
n})\right\\}=\varnothing\ .$

We also infer that

$\text{WF}\left((\mathscr{R}_{A}(F))_{u}^{(n)}\right)\subseteq\text{WF}({\mathcal{R}_{A}}^{\wedge
n})_{M^{n}}\cup\text{WF}^{\prime}({\mathcal{R}_{A}}^{\wedge
n})\circ\text{WF}(F^{(n)}_{\mathcal{R}_{A}u})$

i.e.

$\begin{split}\text{WF}\left((\mathscr{R}_{A}(F))_{u}^{(n)}\right)\subseteq\Big{\\{}&(x_{1},\dots,x_{n},\xi_{1},\dots,\xi_{n})\
|\
(x_{1},y_{1},\dots,x_{n},y_{n},\xi_{1},-\eta_{1},\dots,\xi_{n},-\eta_{n})\in\text{WF}(\mathcal{R}_{A}^{\wedge
n})\\\ &\text{for some}\
(y_{1},\dots,y_{n},\eta_{1},\dots,\eta_{n})\in\text{WF}(F_{\mathcal{R}_{A}u}^{(n)})\Big{\\}}\
.\end{split}$

Using this, we shall prove (5.5) by contradiction. Assume then that
$(x_{1},\dots,x_{n},\xi_{1},\dots,\xi_{n})\in\overline{V_{x_{1}}^{+}}\times\cdots\times\overline{V_{x_{n}}^{+}}$;
then there exists
$(y_{1},\dots,y_{n},\eta_{1},\dots,\eta_{n})\in\text{WF}(F_{\mathcal{R}_{A}u}^{(n)})$
such that

$(x_{1},y_{1},\dots,x_{n},y_{n},\xi_{1},-\eta_{1},\dots,\xi_{n},-\eta_{n})\in\text{WF}({\mathcal{R}_{A}}^{\wedge
n})\ .$

If $i\in I$, then
$(x_{i},y_{i},\xi_{i},-\eta_{i})\in\text{WF}({\mathcal{R}_{A}})$, that is,
either

$(x_{i},y_{i},\xi_{i},-\eta_{i})\in\text{WF}({S_{-}^{G}}^{\oplus})\
\text{with}\ y_{i}\in\mathrm{supp}(\mathscr{A})$

or

$x_{i}=y_{i}\qquad\eta_{i}=\xi_{i}\ .$

In the first case, we would then need to have
$\eta_{i}\in\overline{V^{+}_{y_{i}}}$, as $\eta_{i}$ is the cotangent vector
to a future-directed lightlike geodesic, while in the second one the same
result follows from the equality $\eta_{i}=\xi_{i}$. If $i\notin I$, then

$(x_{i},y_{i},\xi_{i},\eta_{i})=(x_{i},y_{i},0,0)$

i.e. $\eta_{i}=0\in\overline{V^{+}_{y_{i}}}$. Thus, we conclude that
$(y_{1},\dots,y_{n},\eta_{1},\dots,\eta_{n})\in\overline{V^{+}_{y_{1}}}\times\cdots\overline{V^{+}_{y_{n}}}$;
but we reached a contradiction, as this is not possible by the definition of
$F$. The case
$(x_{1},\dots,x_{n},\xi_{1},\dots,\xi_{n})\in\overline{V_{x_{1}}^{-}}\times\cdots\times\overline{V_{x_{n}}^{-}}$
leads to a similar conclusion: indeed, if $i\in I$ then
$(x_{i},y_{i},\xi_{i},-\eta_{i})\in\text{WF}(\mathcal{R}_{A})$, which as
before entails that either

$(x_{i},y_{i},\xi_{i},-\eta_{i})\in\text{WF}({S_{-}^{G}}^{\oplus})\
\text{with}\ y_{i}\in\mathrm{supp}(\mathscr{A})$

or

$x_{i}=y_{i}\qquad\xi_{i}=\eta_{i}\ .$

The first case is not possible, as we require
$(x_{i},\xi_{i})\sim(y_{i},\eta_{i})$ which is not possible if $\xi_{i}$ is
past-directed; thus from the second case we infer that
$(y_{1},\dots,y_{n},\eta_{1},\dots,\eta_{n})\in\overline{V^{-}_{y_{1}}}\times\cdots\overline{V^{-}_{y_{n}}}$,
which is again a contradiction.

We thus have the well-posedness of the map
$\mathscr{R}_{A}\colon\mathcal{A}(S_{G})[\\![\hbar]\\!]\to\mathcal{A}(S)[\\![\hbar]\\!]$.
Recall that $R_{A}$ admits an inverse which is explicitly given by (3.7); it
can be shown that by defining an analogous
$\widehat{\mathscr{R}}_{A}\colon\mathcal{A}(S)[\\![\hbar]\\!]\to\mathcal{A}(S_{G})[\\![\hbar]\\!]$
we reach the same conclusion, and that
$\mathscr{R}_{A}\circ\widehat{\mathscr{R}}_{A}=\mathrm{id}_{\mathcal{A}(S)[\\![\hbar]\\!]}$
and
$\widehat{\mathscr{R}}_{A}\circ\mathscr{R}_{A}=\mathrm{id}_{\mathcal{A}(S_{G})[\\![\hbar]\\!]}$.
Therefore, $\mathscr{R}_{A}$ is a vector space isomorphism.

The fact that $\mathscr{R}_{A}$ is an algebra homomorphism is due to the
following fact: we know that the $\star$-product in
$\mathcal{A}(S)[\\![\hbar]\\!]$ is given by (5.4); given two homogeneous
functionals $F\in\mathcal{A}^{p}(S)$ and $G\in\mathcal{A}^{q}(S)$, the
functional $\Gamma^{n}_{\omega}(G,F)$ appearing in the sum can be formally
written, on an homogeneous element $u_{1}\wedge\cdots\wedge
u_{p+q-2n}\in\wedge^{p+q-2n}\pazocal{E}(S^{\oplus}(M))$ as

$\begin{split}&\Gamma^{n}_{\omega}(G,F)(u_{1}\wedge\cdots\wedge
u_{p+q-2n})=\left(\frac{i}{2}\right)^{n}\sum_{\sigma\in
S_{p+q-2n}}\mathrm{sgn}(\sigma)\int_{M^{2n}}d\mu_{g}(x_{1})d\mu_{g}(y_{1})\cdots
d\mu_{g}(x_{n})d\mu_{g}(y_{n})\\\
&\qquad\left(F^{(n)}_{u_{\sigma(q-n+1)}\wedge\cdots\wedge
u_{\sigma(p+q-2n)}}\right)_{f_{1}\cdots
f_{n}}(y_{1},\dots,y_{n})\left(G^{(n)}_{u_{\sigma(1)}\wedge\cdots\wedge
U_{\sigma(q-n)}}\right)_{g_{1}\cdots g_{n}}(x_{1},\dots,x_{n})\\\
&\qquad\omega_{s_{1}t_{1}}(x_{1},y_{1})\cdots\omega_{s_{n}t_{n}}(x_{n},y_{n})h^{f_{1}s_{1}}h^{t_{1}g_{1}}\cdots
h^{f_{n}s_{n}}h^{t_{n}g_{n}}\ .\end{split}$

We are interested in computing
$\Gamma_{\omega}^{n}(\mathscr{R}_{A}(G),\mathscr{R}_{A}(F))$ with
$F\in\mathcal{A}^{p}(S_{G})$ and $G\in\mathcal{A}^{q}(S_{G})$; this amounts to
substituting the formal integral above with

$\begin{split}&\int_{M^{2n}}d\mu_{g}(\xi_{1})d\mu_{g}(\eta_{1})\cdots
d\mu_{g}(\xi_{n})d\mu_{g}(\eta_{n})d\mu_{g}(x_{1})d\mu_{g}(y_{1})\cdots
d\mu_{g}(x_{n})d\mu_{g}(y_{n})\\\
&\qquad\left(F^{(n)}_{{\mathcal{R}_{A}}^{\wedge
p-n}(u_{\sigma(q-n+1)}\wedge\cdots\wedge
u_{\sigma(p+q-2n)})}\right)_{l_{1}\cdots
l_{n}}(\xi_{1},\dots,\xi_{n})\left(G^{(n)}_{{\mathcal{R}_{A}}^{\wedge
q-n}(u_{\sigma(1)}\wedge\cdots\wedge u_{\sigma(q-n)})}\right)_{k_{1}\cdots
k_{n}}(\eta_{1},\dots,\eta_{n})\\\
&\qquad{r_{A}}_{f_{1}u_{1}}(x_{1},\xi_{1}){r_{A}}_{g_{1}v_{1}}(y_{1},\eta_{1})\cdots{r_{A}}_{f_{n}u_{n}}(x_{n},\xi_{n}){r_{A}}_{g_{n}v_{n}}(y_{n},\eta_{n})h^{l_{1}u_{1}}h^{v_{1}k_{1}}\cdots
h^{l_{n}u_{n}}h^{v_{n}k_{n}}\\\
&\qquad\omega_{s_{1}t_{1}}(x_{1},y_{1})\cdots\omega_{s_{n}t_{n}}(x_{n},y_{n})h^{f_{1}s_{1}}h^{t_{1}g_{1}}\cdots
h^{f_{n}s_{n}}h^{t_{n}g_{n}}\ .\end{split}$

By performing a simple computation one can notice that

$\begin{split}\int_{M^{2}}d\mu_{g}(x_{i})d\mu_{g}(y_{i}){{r}_{A}}_{f_{i}u_{i}}(x_{i},\xi_{i})&{{r}_{A}}_{g_{i}v_{i}}(y_{i},\eta_{i})\omega_{s_{i}t_{i}}(x_{i},y_{i})h^{f_{i}s_{i}}h^{t_{i}g_{i}}\\\
&=\int_{M^{2}}d\mu_{g}(x_{i})d\mu_{g}(y_{i}){{r}_{A}}_{u_{i}f_{i}}^{\ast}(\xi_{i},x_{i}){{r}_{A}}_{v_{i}g_{i}}^{\ast}(\eta_{i},y_{i})\omega_{s_{i}t_{i}}(x_{i},y_{i})h^{f_{i}s_{i}}h^{t_{i}g_{i}}\\\
&={\omega_{G}}_{u_{i}v_{i}}(\xi_{i},\eta_{i})\end{split}$

and therefore we arrive at

$\begin{split}&\int_{M^{2n}}d\mu_{g}(\xi_{1})d\mu_{g}(\eta_{1})\cdots
d\mu_{g}(\xi_{n})d\mu_{g}(\eta_{n})\\\
&\qquad\left(F^{(n)}_{{\mathcal{R}_{A}}^{\wedge
p-n}(u_{\sigma(q-n+1)}\wedge\cdots\wedge
u_{\sigma(p+q-2n)})}\right)_{l_{1}\cdots
l_{n}}(\xi_{1},\dots,\xi_{n})\left(G^{(n)}_{{\mathcal{R}_{A}}^{\wedge
q-n}(u_{\sigma(1)}\wedge\cdots\wedge u_{\sigma(q-n)})}\right)_{k_{1}\cdots
k_{n}}(\eta_{1},\dots,\eta_{n})\\\
&\qquad{\omega_{G}}_{u_{1}v_{1}}(\xi_{1},\eta_{1})\cdots{\omega_{G}}_{u_{n}v_{n}}(\xi_{n},\eta_{n})h^{l_{1}u_{1}}h^{v_{1}k_{1}}\cdots
h^{l_{n}u_{n}}h^{v_{n}k_{n}}\end{split}$

which entails that

$\Gamma^{n}_{\omega}(\mathscr{R}_{A}(G),\mathscr{R}_{A}(F))(u_{1}\wedge\cdots\wedge
u_{p+q-2n})=\mathscr{R}_{A}(\Gamma^{n}_{\omega_{G}}(G,F))(u_{1}\wedge\cdots\wedge
u_{p+q-2n})\ .$

The result can be extended to the whole
$\pazocal{E}^{a}(M^{p+q-2n},S^{\oplus}(M)^{\boxtimes p+q-2n})$ by continuity,
as well as to the whole algebra $\mathcal{A}(S_{G})$ and thus to the whole
algebra of formal power series $\mathcal{A}(S_{G})[\\![\hbar]\\!]$. Therefore,

$\mathscr{R}_{A}(F)\star\mathscr{R}_{A}(H)=\mathscr{R}_{A}(F\star_{G}H)$

and $\mathscr{R}_{A}$ is an algebra homomorphism as required.

As far as the behaviour of the classical Møller map with respect to
conjugation is concerned, let us recall that on
$\mathcal{A}(S)[\\![\hbar]\\!]$ and $\mathcal{A}(S_{G})[\\![\hbar]\\!]$ the
conjugation map is given by the natural extension of (5.1) (with the
appropriate conjugation map, that is, with either
$C\colon\pazocal{E}(\overline{S(M)})\to\pazocal{E}(S(M))$ or
$C_{G}\colon\pazocal{E}(\overline{S_{G}(M)})\to\pazocal{E}(S_{G}(M))$) to
formal power series.

Given $F\in\mathcal{A}(S_{G})\subseteq\mathcal{A}(S_{G})[\\![\hbar]\\!]$ and
$u\in\mathcal{C}(S^{\oplus}(M))$ we thus have that

$\begin{split}(\mathscr{R}_{A}(F))^{\ast}(u)&=\overline{(\mathscr{R}_{A}(F))(u^{\ast})}=\sum_{p\in\mathbb{N}}\overline{\langle
F_{p},{\mathcal{R}_{A}}^{\wedge p}u_{p}^{\ast}\rangle}\ .\end{split}$

Now, if $u_{p}=u_{i_{1}}\wedge\cdots\wedge u_{i_{p}}$,
$u_{i_{j}}\in\pazocal{E}(S^{\oplus}(M))$ we have that

${\mathcal{R}_{A}}^{\wedge
p}(u_{p}^{\ast})=\mathcal{R}_{A}u_{i_{p}}^{\ast}\wedge\cdots\wedge\mathcal{R}_{A}u_{i_{1}}^{\ast}=(\mathcal{R}_{A}u_{i_{p}})^{\ast}\wedge\cdots\wedge(\mathcal{R}_{A}u_{i_{1}})^{\ast}=\left({\mathcal{R}_{A}}^{\wedge
p}(u_{1}\wedge\cdots\wedge u_{p})\right)^{\ast}\ .$

By continuity then it holds that

$\sum_{p\in\mathbb{N}}\overline{\langle
F_{p},{\mathcal{R}_{A}}^{\wedge^{p}}u_{p}^{\ast}\rangle}=\sum_{p\in\mathbb{N}}\overline{\langle
F_{p},({\mathcal{R}_{A}}^{\wedge
p}u_{p})^{\ast}\rangle}=(\mathscr{R}_{A}(F^{\ast}))(u)\ .$

Therefore,
$\mathscr{R}_{A}\colon\mathcal{A}(S_{G}(M))[[\hbar]]\to\mathcal{A}(S(M))[[\hbar]]$
is a well-defined algebraic $\ast$-isomorphism. For what concerns the
topological part, it is simple to understand that the sequential completeness
of Hörmander topology is satisfied, since pointwise convergence holds term by
term for the formal power series and that the wave front set condition is as
well satisfied by the construction seen before. ∎

##  6\. Conclusions and outlook

Our classic treatment of fermions in external backgrounds has proceeded with
the aim at establishing the most general framework possible for the passage to
the quantum case, having in mind perturbation theory hence in the language of
formal power series. In doing so we have, however, taken shortcuts that is
worth mentioning again. Two of them are particularly important: the first is
that we have dealt with simply connected spacetimes, which rules out
topological effects (_i.e._ inequivalent spinor structures [28] and Aharonov-
Bohm like effects, for which see, _e.g._ , [12, 40]), the second is our
simplifying assumption about the compactness of the support of the gauge
potentials, which rules out Coulomb potentials. Both assumptions can be
relaxed, but to keep this paper into a reasonable length we postpone any
further discussion. However, notice that for the first problem, the passage to
Fredenhagen’s universal algebra [19] may help to solve the issue and that, as
far as the second is concerned, it is exactly due to the compactness of the
_spatial_ support of the potentials that one rules out secular effects in
perturbation theory [21].

In this paper we have privileged the _pointwise_ treatment of the geometric
structures (sections, potentials _etc_.) to guarantee a detailed and
unambiguous discussion of their features. In particular, our most interesting
result has been to show how the _classical_ Møller maps are algebraic and
topological isomorphisms of the charged and uncharged microcausal fermionic
algebras, as formal power series. Here, the use of wave front sets was
essential. As for a possible next step, once the appropriate interactions are
introduced, one would construct the respective _quantum_ Møller maps (see,
_e.g._ , [14]) and proceed to the explicit computation of physical effects.
One of the most ambitious aims would be, for instance, to compute the Lamb
Shift for hydrogenoid atoms [16] from first principles, avoiding _ad hoc_
assumptions and rigorously controlling eventual approximations.

## Acknowledgement

It is a pleasure to thank Nicola Pinamonti for his interest in our research.

## References

  * [1] Christian Bär  “Green-hyperbolic operators in globally hyprbolic spacetimes”  In _Communications in Mathematical Physics_ 333, 2015, pp. 1585–1615  DOI: https://doi.org/10.1007/s00220-014-2097-7
  * [2] Christian Bär, Paul Gauduchon and Andrei Moroianu  “Generalized cylinders in semi-Riemannian and spin geometry”  In _Mathematische Zeitschrift_ 249.3, 2005, pp. 545–580  DOI: 10.1007/s00209-004-0718-0
  * [3] Christian Bär, Nicolas Ginoux and Frank Pfäffle  “Wave Equations on Lorentzian Manifolds and Quantization”  European Mathematical Society, 2007  DOI: 10.4171/037
  * [4] Christian Brouder, Nguyen Viet Dang and Frédéric Hélein  “A smooth introduction to the wavefront set”  In _Journal of Physics A: Mathematical and Theoretical_ 47.44 IOP Publishing, 2014, pp. 443001  DOI: 10.1088/1751-8113/47/44/443001
  * [5] Christian Brouder, Nguyen Viet Dang, Camille Laurent-Gengoux and Kasia Rejzner  “Properties of field functionals and characterization of local functionals”  In _Journal of Mathematical Physics_ 59.2, 2018, pp. 023508  DOI: 10.1063/1.4998323
  * [6] Romeo Brunetti, Klaus Fredenhagen and Pedro Lauridsen Ribeiro  “Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields”  In _Communications in Mathematical Physics_ 368.2, 2019, pp. 519–584  DOI: 10.1007/s00220-019-03454-z
  * [7] Romeo Brunetti, Klaus Fredenhagen and Rainer Verch  “The Generally Covariant Locality Principle - A New Paradigm for Local Quantum Field Theory”  In _Communications in Mathematical Physics_ 237.1, 2003, pp. 31–68  DOI: 10.1007/s00220-003-0815-7
  * [8] Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen and Katharzina Rejzner  “C*-algebraic approach to interacting quantum field theory: inclusion of Fermi fields”  In _Letters in Mathematical Physics_ , 2022, pp. 112:101  DOI: https://doi.org/10.1007/s11005-022-01590-7
  * [9] Claudio D’Antoni and Stefan Hollands  “Nuclearity, Local Quasiequivalence and Split Property for Dirac Quantum Fields in Curved Spacetime”  In _Communications in Mathematical Physics_ 261.2, 2006, pp. 133–159  DOI: 10.1007/s00220-005-1398-2
  * [10] Claudio Dappiaggi, Thomas-Paul Hack and Nicola Pinamonti  “Approximate KMS states for scalar and spinor fields in Friedmann-Robertson-Walker spacetimes”  In _Annales Henri Poincaré_ 12.10, 2011, pp. 1449–1489  DOI: 10.1007/s00023-011-0111-6
  * [11] Claudio Dappiaggi, Thomas-Paul Hack and Nicola Pinamonti  “The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor”  In _Reviews in Mathematical Physics_ 21.10, 2009, pp. 1241–1312  DOI: 10.1142/S0129055X09003864
  * [12] Claudio Dappiaggi, Giuseppe Ruzzi and Ezio Vasselli  “Aharonov-Bohm superselection sectors”  In _Letters in Mathematical Physics_ 110, 2020, pp. 3243–3278  DOI: https://doi.org/10.1007/s11005-020-01335-4
  * [13] Jonathan Dimock  “Quantized Electromagnetic Field on a Manifold”  In _Reviews in Mathematical Physics_ 04.02, 1992, pp. 223–233  DOI: 10.1142/S0129055X92000078
  * [14] Nicolò Drago, Thomas-Paul Hack and Nicola Pinamonti  “The Generalised Principle of Perturbative Agreement and the Thermal Mass”  In _Annales Henri Poincaré_ 18.3, 2017, pp. 807–868  DOI: 10.1007/s00023-016-0521-6
  * [15] Johannes J. Duistermaat and Lars Hörmander  “Fourier integral operators. II”  In _Acta Mathematica_ 128 Institut Mittag-Leffler, 1972, pp. 183–269  DOI: 10.1007/BF02392165
  * [16] Michael I. Eides, Howard Grotch and Valery A. Shelyuto  “Theory of light hydrogenlike atoms”  In _Physics Report_ 342, 2001, pp. 63–261  DOI: https://doi.org/10.1016/S0370-1573(00)00077-6
  * [17] A. Fedotov et al.  “Advances in QED with intense background fields”  In _Physics Report_ 1010, 2023, pp. 1–138  DOI: https://doi.org/10.1016/j.physrep.2023.01.003
  * [18] Christopher J. Fewster and Rainer Verch  “A Quantum Weak Energy Inequality for Dirac fields in curved spacetime”  In _Communications in Mathematical Physics_ 225, 2002, pp. 331–359  DOI: 10.1007/s002200100584
  * [19] Klaus Fredenhagen  “Global observables in local quantum physics”  In _Quantum and Non-Commutative Analysis: Past, Present and Future Perspectives_ Kluwer Academi Publishers, 1993, pp. 41–51
  * [20] Markus B. Fröb and Jochen Zahn  “Trace anomaly for chiral fermions via Hadamard subtraction”  In _Journal of High Energy Physics_ 2019, 2019, pp. 223  DOI: 10.1007/JHEP10(2019)223
  * [21] Stefano Galanda, Nicola Pinamonti and Leonardo Sangaletti  “Secular growths and their relations to equilibrium states in perturbative QFT”, 2023  DOI: 10.48550/arXiv.2312.00556
  * [22] Robert Geroch  “Spinor Structure of Space‐Times in General Relativity. I”  In _Journal of Mathematical Physics_ 9.11, 1968, pp. 1739–1744  DOI: 10.1063/1.1664507
  * [23] Nicolas Ginoux  “Linear Wave Equations”  In _Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2009, pp. 59–84  DOI: 10.1007/978-3-642-02780-2˙3
  * [24] Mark J.. Hamilton  “Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics”  Springer International Publishing, 2017  DOI: 10.1007/978-3-319-68439-0
  * [25] S.. Hawking and G… Ellis  “The Large Scale Structure of Space-Time”  Cambridge University Press, 1973  DOI: 10.1017/CBO9780511524646
  * [26] Stefan Hollands  “The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes”  In _Communications in Mathematical Physics_ 216.2, 2001, pp. 635–661  DOI: 10.1007/s002200000350
  * [27] Lars Hörmander  “The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis”  Springer Berlin Heidelberg, 1998  DOI: 10.1007/978-3-642-96750-4
  * [28] Christopher J. Isham  “Spinor Fields in Four Dimensional Space-Time”  In _Proc. R. Soc. Lond. A_ 364, 1978, pp. 591–599
  * [29] Shoshichi Kobayashi and Katsumi Nomizu  “Foundations of Differential Geometry, Volume 1”  Interscience Publishers New York - London, 1963
  * [30] H. Lawson and Marie-Louise Michelsohn  “Spin Geometry”  Princeton University Press, 1990
  * [31] Simone Murro and Daniele Volpe  “Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds”  In _Annals of Global Analysis and Geometry_ 59.1, 2021, pp. 1–25  DOI: 10.1007/s10455-020-09739-0
  * [32] Gregory L. Naber  “Topology, Geometry and Gauge fields: Foundations”  Springer New York, 2011  DOI: 10.1007/978-1-4419-7254-5
  * [33] Liviu I Nicolaescu  “Lectures on the Geometry of Manifolds”  World Scientific, 2020  DOI: 10.1142/11680
  * [34] Marek J. Radzikowski  “Micro-local approach to the Hadamard condition in quantum field theory on curved space-time”  In _Communications in Mathematical Physics_ 179.3 Springer, 1996, pp. 529–553  DOI: 10.1007/BF02100096
  * [35] Katarzyna Rejzner  “Fermionic Fields in the Functional Approach to Classical Field Theory”  In _Reviews in Mathematical Physics_ 23.09, 2011, pp. 1009–1033  DOI: 10.1142/S0129055X11004503
  * [36] Ko Sanders  “The Locally Covariant Dirac Field”  In _Reviews in Mathematical Physics_ 22.04, 2010, pp. 381–430  DOI: 10.1142/S0129055X10003990
  * [37] Jan Schlemmer and Jochen Zahn  “The current density in quantum electrodynamics in external potentials”  In _Annals of Physics_ 359, 2015, pp. 31–45  DOI: 10.1016/j.aop.2015.04.006
  * [38] Alexander Stromahier  “Microlocal Analysis”  In _Quantum Field Theories in Curved Spacetimes_ , Lecture Notes in Physics, vol. 786  Berlin Heidelberg: Springer-Verlag, 2009, pp. 85–127  DOI: https://doi.org/10.1007/978-3-642-02780-2˙4
  * [39] Loring W. Tu  “Differential Geometry: Connections, Curvature, and Characteristic Classes”  Springer International Publishing, 2017  DOI: 10.1007/978-3-319-55084-8
  * [40] Ezio Vasselli  “Background potentials and superselection sectors”  In _Journal of Geometry and Physics_ 139, 2019, pp. 139–148  DOI: https://doi.org/10.1016/j.geomphys.2019.02.001
  * [41] Jochen Zahn  “The current density in quantum electrodynamics in time-dependent external potentials and the Schwinger effect”  In _Journal of Physics A: Mathematical and Theoretical_ 48, 2015, pp. 475402  DOI: 10.1088/1751-8113/48/47/475402
  * [42] Jochen Zahn  “The Renormalized Locally Covariant Dirac Field”  In _Reviews in Mathematical Physics_ 26.01, 2014, pp. 1330012  DOI: 10.1142/S0129055X13300124