Møller Maps for Dirac Fields in External Backgrounds

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 already present in various places in the literature. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions $d\ge 4$ and to external fields modelled with trivial principal bundles. In particular, we construct the classical Møller maps intertwining the configuration spaces of charged and uncharged fermions, and we show some of its properties in the case of a U(1) gauge charge. 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.

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., [1,2,3,4,5,6,7,8,9,10,11,12,13]). 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 preferred 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., [14, 15]). 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 [16], and technically, instead of Fourier transformations, uses its modern improvement named micro-local analysis [17], in particular wave front sets [18, 19]. There have been several recent interesting results that point towards the validity of this claim. For instance, Fröb and Zahn [10] 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 several bundles over semi-Riemannian spaces, and prove that the Møller maps are isomorphisms between the charged and uncharged spinor bundles. Then, 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., [20]). 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 view which has the merit of being rather explicit. The resulting formalism has 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 [21,22,23] and [24] 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:

(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., [25, 26]);

(ii):

we also assume that $\text {dim}(M)\ge 4$: this entails that there exists a universal covering homomorphism $\xi _0:\text {Spin}^0_{r,1} \rightarrow \text {SO}_{r,1}^0$ between the identity component of the $\text {Spin}$ group $\text {Spin}_{r,1}$ and the identity component of the signature (r, 1) of the special orthogonal group [24, Proposition 12.1.41].

On the manifold M modelling our spacetime, we shall consider fiber bundles, i.e., quadruples $(B, M, \pi , F)$ where $\pi :B \rightarrow M$ 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 $\{\phi _\alpha :\pi ^{-1}(U_\alpha ) \rightarrow U_\alpha \times F\}_{\alpha \in A}$ of diffeomorphisms, called trivialization of the bundle, such that

$$\begin{aligned} \pi _1 \circ \phi _\alpha = \pi |_{\pi ^{-1}(U_\alpha )} \ \text {for every} \ \alpha \in A\ . \end{aligned}$$

(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 why F is called typical fiber. We can consider the transition functions of the bundle, that is, the maps

$$\begin{aligned} \begin{aligned} \phi _{\alpha \beta }:(U_\alpha \cap U_\beta )\times F&\rightarrow (U_\alpha \cap U_\beta )\times F\\ (p, f)\qquad&\mapsto \phi _\alpha \left( \phi _\beta ^{-1}(p, f)\right) \ . \end{aligned} \end{aligned}$$

By the condition (2.1), we have that $\phi _{\alpha \beta }(p, f) = (p, g_{\alpha \beta }(p)(f))$ for some $g_{\alpha \beta }:U_\alpha \cap U_\beta \rightarrow \text {Aut}(F)$; 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:

(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 $\alpha , \beta \in A$, $p\in U_\alpha \cap U_\beta$;

(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 $r_g:P_G \rightarrow P_G$ and whose trivialization $\{\phi _\alpha :\pi ^{-1}(U_\alpha ) \rightarrow U_\alpha \times G\}$ consists of right G-equivariant maps. This entails that the cocycle $\{g_{\alpha \beta }:U_\alpha \cap U_\beta \rightarrow \text {Aut}(G)\}$ consists of left-translations [21, Lemma 27.7].

Given a principal G-bundle $\pi :P_G \rightarrow M$ and a representation $\rho :G \rightarrow \text {GL}(V)$ of the group on a finite-dimensional vector space V, we can construct the associated vector bundle $\pi :P_G \times _\rho V \rightarrow M$ with typical fiber V. [21, Theorem 31.9] shows that there exist linear isomorphisms

$$\begin{aligned} \begin{aligned}&{\cdot }^\flat :\Omega _\rho ^q(P_G, V)\rightarrow \Gamma \left( \wedge ^q T^*M \otimes P_G\times _\rho V\right) \\ {}&{\cdot }^\sharp :\Gamma \left( \wedge ^q T^*M \otimes P_G\times _\rho V\right) \rightarrow \Omega _\rho ^q(P_G, V) \end{aligned} \end{aligned}$$

(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 [27, 28]. These connection 1-forms can be used to induce an exterior covariant differentiation $D:\Omega _\rho ^q(P_G, V)\rightarrow \Omega _\rho ^{q+1}(P_G ,V)$,

$$\begin{aligned} D\varphi \overset{\cdot }{=}d\varphi + \omega \cdot _\rho \varphi \end{aligned}$$

(2.3)

where

$$\begin{aligned} (\omega \cdot _\rho \varphi )_{p}(v_1, \ldots , v_{q+1}) = \frac{1}{q!} \sum _{\sigma \in S_{q+1}}\text {sgn}(\sigma )\rho _*(\omega _p(v_{\sigma (1)}))\varphi _p(v_{\sigma (2)}, \ldots , v_{\sigma (q+1)})\ . \end{aligned}$$

We can then use the exterior covariant derivative to endow the associated vector bundle $P_G \times _\rho V$ with a connection $\nabla :{\mathfrak {X}}(M)\times \Gamma (P_G\times _\rho V)\rightarrow \Gamma (P_G\times _\rho V)$,

$$\begin{aligned} \nabla _v s \overset{\cdot }{=}(D(s^\sharp ))^\flat (v)\ . \end{aligned}$$

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

$$\begin{aligned} (\nabla _v s)(p) = \left[ \sigma (p), v(p) \left( s^\sharp \circ \sigma \right) +\rho _*\Big (\omega _{\sigma (p)}\big ({\sigma _*}_p(v_p)\big )\Big )s^\sharp (\sigma (p))\right] \end{aligned}$$

(2.4)

where $\sigma :U \rightarrow {P_G}_U$ is a local section of the principal G-bundle $P_G$, usually deemed local gauge choice. The pullback $\sigma ^*\omega \in \Omega ^1(U, {\mathfrak {g}})$ 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 $C:{\overline{V}}\rightarrow V$, and it can be used to induce, starting from a linear map $L:V \rightarrow V$, a linear map ${\overline{L}}:{\overline{V}}\rightarrow {\overline{V}}$ by defining

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

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.

$$\begin{aligned} {\overline{L}}^i_j = \overline{L^i_j}\ . \end{aligned}$$

If V is endowed with a sesquilinear form, then one can show that ${\overline{V}}\simeq V^*$. These facts can be extended to the case of complex vector bundles in the following way: first of all, consider a vector bundle $E \overset{\pi }{\rightarrow }\ M$ 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 ${\overline{E}}\overset{\pi }{\rightarrow }\ M$. The conjugation map $C:{\overline{V}}\rightarrow V$ can be extended to a vector bundle anti-isomorphism $C:{\overline{E}}\rightarrow E$, and if E is endowed with a hermitian metric h, then

• (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 $h_p:E_p \times {\overline{E}}_p\rightarrow {\mathbb {C}}$;

• (ii) we have an isomorphism ${\overline{E}}\simeq E^*$.

Let us now consider the tangent bundle $\pi :TM \rightarrow M$ 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 $\pi _P :P_{\text {SO}_{r,1}^0}(M)\rightarrow M$ 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 $\text {Spin}_{r,1}^0$-bundle $\pi _{P'}:P_{\text {Spin}_{r,1}^0}(M)\rightarrow M$ coupled with a 2-sheeted covering

$$\begin{aligned} \xi :P_{\text {Spin}^0_{r,1}}\rightarrow P_{\text {SO}_{r,1}^0} \end{aligned}$$

(2.5)

such that $\xi \left( r_g(p)\right) = r_{\xi _0(g)}\left( \xi (p)\right)$ for every $p\in P_{\text {Spin}_{r,1}^0}$, $g\in \text {Spin}_{r,1}^0$, with $\xi _0:\text {Spin}_{r,1}^0\rightarrow \text {SO}_{r,1}^0$ being the universal covering map. This entails in particular that the cocycle $\{g_{\alpha \beta }:U_\alpha \cap U_\beta \rightarrow \text {Aut}(\text {SO}_{r,1}^0)\}$ of the principal $\text {SO}_{r,1}^0$-bundle is given by $\{\xi _0(s_{\alpha \beta })\}$, where $\{s_{\alpha \beta }:U_\alpha \cap U_\beta \rightarrow \text {Aut}(\text {Spin}_{r,1}^0)\}$ is a cocycle for the principal $\text {Spin}_{r,1}^0$-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 $\omega ^s\in \Omega ^1(P_{\text {Spin}_{r,1}^0}, \mathfrak {spin}_{r,1}^0)$, 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 $\Delta _{r,1}^{{\mathbb {C}}}:\text {Spin}_{r,1}^0\rightarrow \text {GL}(V, {\mathbb {C}})$ which is irreducible in the odd dimensional case and reducible in the even dimensional one [22, §I, Proposition 5.15]. In both cases, the representation descends from a complex representation $\rho :{\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}\rightarrow \text {End}(V, {\mathbb {C}})$ of the Clifford algebra ${\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$. In [22, §I, Theorem 5.7] it is proved 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:

(i):

given a complex Spin representation $\Delta _{r,1}^{{\mathbb {C}}}:\text {Spin}_{r,1}^0\rightarrow \text {GL}(V, {\mathbb {C}})$, we can consider the complex spinor bundle

$$\begin{aligned} S(M)\overset{\cdot }{=}P_{\text {Spin}_{r,1}^0}\times _{\Delta _{r,1}^{{\mathbb {C}}}} V \end{aligned}$$

(2.6)

(ii):

we can also consider the Clifford algebra bundle

$$\begin{aligned} {\mathscr {C}}\hspace{-1.5pt}\ell (M)\overset{\cdot }{=}P_{\text {SO}_{r,1}^0}\times _{c\hspace{-0.5pt}\ell }{\mathscr {C}}\hspace{-1.5pt}\ell _{r,1} \end{aligned}$$

(2.7)

where $c\hspace{-0.5pt}\ell :\text {SO}_{r,1}^0\rightarrow \text {Aut}({\mathscr {C}}\hspace{-1.5pt}\ell _{r,1})$ denotes the unique extension to the Clifford algebra ${\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$ 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 ${\mathscr {C}}\hspace{-1.5pt}\ell (M)$ at $p\in M$ is isomorphic to ${\mathscr {C}}\hspace{-1.5pt}\ell (T_p^*M, g_p)$.

These two bundles are related by means of the Clifford module multiplication

$$\begin{aligned} \mu :{\mathscr {C}}\hspace{-1.5pt}\ell (M)\otimes S(M)\rightarrow S(M) \end{aligned}$$

which explicitly uses the fact that $\Delta _{r,1}^{{\mathbb {C}}}$ descends from an algebra representation of ${\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$.

Thanks to the spin structure, this map can then be extended to yield a Clifford module multiplication

$$\begin{aligned} \cdot _{\Gamma }:{\mathfrak {X}}( M)\otimes \Gamma (S(M))\rightarrow \Gamma (S(M)) \end{aligned}$$

(2.8)

between vector fields and sections of the spinor bundle.

We shall denote with

$$\begin{aligned} \nabla ^s:{\mathfrak {X}}(M)\otimes \Gamma (S(M))\rightarrow \Gamma (S(M)) \end{aligned}$$

(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

$$\begin{aligned} \nabla _u^s(v \cdot _\Gamma s) = (\nabla _u v)\cdot _\Gamma s + v \cdot _\Gamma \nabla _u^s s\ . \end{aligned}$$

(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

$$\begin{aligned} \begin{aligned} {{\textbf {D}}}:\Gamma (S(M))&\rightarrow \Gamma (S(M))\\ s\qquad&\mapsto \quad {{\textbf {D}}}(s)\overset{\cdot }{=}i\left( \cdot _\Gamma (\nabla ^s(s))\right) \ . \end{aligned} \end{aligned}$$

(2.11)

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

$$\begin{aligned} {{\textbf {D}}}(s)(p) = ie^j(p) \cdot _\Gamma \left[ \sigma (p),e_j(p)\left( s^\sharp \circ \sigma \right) + {\Delta _{r,1}^{{\mathbb {C}}}}_*\left( \omega _{\sigma (p)}^s(\sigma _{*_p}e_j(p))\right) s^\sharp (\sigma (p))\right] \nonumber \\ \end{aligned}$$

(2.12)

where we use Einstein’s convention.

Remark 2.1

The local expression above can be written even more explicitly as

$$\begin{aligned} {{\textbf {D}}}(s)(p) = ie^j(p) \cdot _\Gamma \left[ \sigma (p),e_j(p)\left( s^\sharp \circ \sigma \right) + \frac{1}{4}\sum _{k,l} \Gamma _{kj}^l e^k(p) \cdot _V e_l(p) \cdot _V s^\sharp (\sigma (p))\right] \end{aligned}$$

where $\Gamma _{jk}^l$ denote the Christoffel symbols of the Levi–Civita connection on M inducing the spin connection $\omega ^s$ via pullback. In the following, we present the derivation of the explicit form of the Dirac operator in three different spacetimes.

(a):

Let us consider (flat) Minkowski spacetime $(M,g) = ({\mathbb {R}}^4, \eta )$ and the embedding of ${\mathfrak {X}}({\mathbb {R}}^4)$ into $\Gamma ({\mathscr {C}}\hspace{-1.5pt}\ell ({\mathbb {R}}^4))$ given by $e_j \mapsto \gamma _j$,$\gamma _j (p) \equiv \gamma _j$ where $\{\gamma _j\}_{1\le j \le 4}$ are the usual Gamma matrices in the Dirac representation. By choosing the global pseudo-orthonormal frame $\{\partial _i\}_{1\le i \le 4}$ one can write, for a function $s\in \Gamma (S({\mathbb {R}}^4)) \simeq C^\infty ({\mathbb {R}}^4, {\mathbb {C}}^4)$

$$\begin{aligned} {{\textbf {D}}}(s) = i \gamma ^j \partial _j s \end{aligned}$$

which is the usual expression of the (massless) Dirac operator.

(b):

A more interesting example is that of the Dirac operator in Schwarzschild spacetime,

$$\begin{aligned}{} & {} M = {\mathbb {R}}\times (r_g, +\infty )\times {\mathbb {S}}^1 \\ {}{} & {} g = -\left( 1-\frac{r_g}{r}\right) dt\otimes dt + \left( 1-\frac{r_g}{r}\right) ^{-1}dr \otimes dr + r^2g_{{\mathbb {S}}^2} \end{aligned}$$

To carry out the computations, one considers a (in this case global) pseudo-orthonormal frame $\{e_j\}_{1\le j \le 4}$,

$$\begin{aligned} e_1 = \left( 1-\frac{r_g}{r}\right) ^{-1/2}\partial _t \quad e_2 = \left( 1-\frac{r_g}{r}\right) ^{1/2} \partial _r \quad e_3 = \frac{1}{r}\partial _\theta \quad e_4 = \frac{1}{r\sin (\theta )}\partial _\varphi \end{aligned}$$

and the associated embedding $e_j\mapsto \gamma _j$ where, as before, $\{\gamma _j\}_{1\le j \le 4}$ are the Gamma matrices in the Dirac representation. In this non-holonomic, pseudo-orthornormal frame the Christoffel symbols are given by

$$\begin{aligned} \Gamma ^i_{jk} = \frac{1}{2}\eta ^{im}\left( c_{mjk}+c_{mkj}-c_{jkm}\right) \end{aligned}$$

where $\eta$ is the flat metric and

$$\begin{aligned} {[}e_i, e_j] = c^{m}_{jk}e_m, \quad c_{jkp}=\eta _{pm}c^{m}_{jk} \end{aligned}$$

One can also exploit the properties of Christoffel symbols and of the Clifford module multiplication to write

$$\begin{aligned} \frac{1}{4}\sum _{k,l}\Gamma _{mkj} e^k (p) \cdot _V e^j(p) = \frac{1}{8}\sum _{k,l} \Gamma _{mkj}[\gamma ^k, \gamma ^j] \end{aligned}$$

A lengthy but straightforward computation then leads to

$$\begin{aligned} \begin{aligned} {{\textbf {D}}}(s)&= i \gamma ^1\left( 1{-}\frac{r_g}{r}\right) ^{{-}1/2}\partial _t s {+}i\gamma ^2\left( 1{-}\frac{r_g}{r}\right) ^{1/2}\left( \partial _r s {-}\frac{r_g}{4r^2}\left( 1{-}\frac{r_g}{r}\right) ^{-1} s{-}\frac{1}{r} s\right) \\&\quad + i\gamma ^3\left( \frac{1}{r}\partial _\theta s -\frac{1}{2}\frac{\cot (\theta )}{r}s\right) +i\gamma ^4\frac{1}{r\sin (\theta )}\partial _\varphi s\ . \end{aligned} \end{aligned}$$

(c):

Another interesting example is that of Friedman–Lemaitre–Robertson–Walker spacetime,

$$\begin{aligned} M= I \times J \times {\mathbb {S}}^2 \qquad g = S(t)^2\left( -dt\otimes dt + dr \otimes dr + f(r)^2 g_{{\mathbb {S}}^2}\right) \ . \end{aligned}$$

Proceeding in the same way as in the Schwarzschild case, a simple computation leads to

$$\begin{aligned} \begin{aligned} {{\textbf {D}}}(s)&= i \gamma ^1\left( \frac{1}{S(t)}\partial _t s - \frac{3}{2}\frac{{\dot{S}}(t)}{S(t)^2}s\right) + i\gamma ^2\left( \frac{1}{S(t)}\partial _r s - \frac{f'(r)}{S(t)f(r)}s\right) \\&\quad + i\gamma ^3\left( \frac{1}{S(t)f(r)}\partial _\theta s - \frac{\cot (\theta )}{S(t)f(r)}s\right) + i\gamma ^4 \frac{1}{S(t)f(r)\sin (\theta )}\partial _\varphi s\ . \end{aligned} \end{aligned}$$

As one can see from (2.12), the Dirac operator is a first-order partial differential operator whose principal symbol $\sigma ({{\textbf {D}}})$ is given pointwise by

$$\begin{aligned} \sigma ({{\textbf {D}}})\left( p, \xi _p\right) (s_p) = \mu |_p(\xi _p, s_p)\ . \end{aligned}$$

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

(2.13)

i.e. , and [26], where $J^{\pm }(R)$ is, respectively, the causal future/past of the region $R\subset M$. These operators can be extended to continuous linear maps

where the subscripts $\text {fc}$ and $\text {pc}$ denote sections with future-compact and past-compact support respectively; we recall here that a subset $S\subseteq M$ is past-compact if ${\overline{S}}\cap J^-(p)$ is compact for every $p\in M$ (future-compact are defined in an analogous way). We remind, for later purposes, that we can also define the so-called causal Green operator (or, more simply, propagator) by posing .

Remark 2.2

In the example case of Minkowski spacetime $({\mathbb {R}}^4, \eta )$ previously examined, the advanced and retarded Green operators for the (massive) Dirac operator can be written explicitly by using the fact that

where $\Delta _\pm$ denote the advanced and retarded propagators of the Klein-Gordon operator, whose integral kernel can be written, in the sense of distributions, as

$$\begin{aligned} \Delta _\pm (x-y)=\frac{1}{(2\pi )^4}\lim _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}^4}\frac{e^{ik_0(x^0-y^0)+i({{\textbf {k}}}, {{\textbf {x}}}-{{\textbf {y}}})}}{-(k_0\mp i\varepsilon )^2 + \left\Vert {{\textbf {k}}}\right\Vert ^2+m^2}\, dk_0 d{{\textbf {k}}}\ . \end{aligned}$$

Then

Explicit expressions for the advanced and retarded propagators in the more interesting case of de Sitter spacetime can be found in [29].

By [24, Proposition 12.1.65], we know that S(M) admits a Hermitian metric such that the Clifford multiplication by $\alpha \in \Gamma (T^*M)$ 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 [8], the coupling of a Dirac field with an external gauge field exploits the notion of direct product bundle, which we recall here:

Definition 2.3

Let $\pi _P:P \rightarrow M$ and $\pi _Q :Q \rightarrow M$ 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 $\Delta =\{(x,x)\in M \times M, \ x\in M\}$ and the inclusion map given by $i:\Delta {\hookrightarrow }M\times M$, we can then consider the pullback bundle

$$\begin{aligned} P+Q \overset{\cdot }{=}i^*(P\times Q) = \left\{ (x, p,q)\in \Delta \times (P \times Q) \ \text {s.t.} \ (x,x)=(\pi _P(p),\pi _Q(q))\right\} \ . \end{aligned}$$

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

$$\begin{aligned} f_P :P \times Q \rightarrow P \qquad f_Q :P \times Q \rightarrow Q \end{aligned}$$

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

$$\begin{aligned} f_P\left( r_{(g,h)}(p,q)\right) = r_g p \qquad f_Q\left( r_{(g,h)}(p,q)\right) = r_h q \ . \end{aligned}$$

If the two principal bundles in Definition 2.3 are endowed with the connections $\omega ^P \in \Omega ^1(P, \mathfrak {g})$ and $\omega ^Q\in \Omega ^1(Q, \mathfrak {h})$ respectively, then by [30, Proposition 6.3] we have that the 1-form

$$\begin{aligned} \omega \overset{\cdot }{=}f_P^*\omega ^P \oplus f_Q^*\omega ^Q \end{aligned}$$

(2.14)

is an Ehresmann connection on $P+Q$.

Let us now consider two representations

$$\begin{aligned} \rho _1:G \rightarrow \text {GL}(V) \qquad \rho _2:H \rightarrow \text {GL}(V) \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \rho :G\times H&\rightarrow \text {GL}(V)\\ (g,h)&\mapsto \rho \left( g,h\right) \overset{\cdot }{=}\rho _1(g)\rho _2(h) \end{aligned} \end{aligned}$$

(2.15)

whose adjoint representation is given by

$$\begin{aligned} \begin{aligned} \rho _*:{\mathfrak {g}}\oplus {\mathfrak {h}}&\rightarrow \qquad \text {End}(V)\\ ({{\textbf {g}}}, {{\textbf {h}}})&\mapsto {\rho _1}_*({{\textbf {g}}})+{\rho _2}_*({{\textbf {h}}}) \end{aligned} \end{aligned}$$

and consider the associated bundle

$$\begin{aligned} (P+Q)\times _\rho V \end{aligned}$$

which will then be endowed with a covariant derivative.

In our case of interest we will deal with a principal $\text {Spin}_{r,1}^0$-bundle $\pi _s:P_{\text {Spin}_{r,1}^0}\rightarrow M$ coupled with a principal G-bundle $\pi _g:P_G \rightarrow M$, where G is a compact Lie group which admits a representation $\rho _G:G\rightarrow \text {GL}(V)$ such that

$$\begin{aligned} \Delta _{r,1}^{{\mathbb {C}}}(s)\rho _G(g) = \rho _G(g) \Delta _{r,1}^{{\mathbb {C}}}(s) \ \text {for every} \ s\in \text {Spin}_{r,1}^0, \ g\in G\ . \end{aligned}$$

By considering the direct product bundle $P_{\text {Spin}_{r,1}^0} + P_G$ endowed with the connection given by (2.14) as well as the representation given by (2.15) we then can construct the associated vector bundle

$$\begin{aligned} S_G(M) \overset{\cdot }{=}(P_{\text {Spin}_{r,1}^0} + P_G) \times _\rho V \end{aligned}$$

which we shall call charged spinor bundle. This can be endowed with the covariant derivative

$$\begin{aligned} \nabla ^{s,G}:{\mathfrak {X}}(M)\otimes \Gamma (S_G(M))\rightarrow \Gamma (S_G(M))) \end{aligned}$$

locally given by

$$\begin{aligned} \begin{aligned} (\nabla ^{s,G}_v s)(p)&= \left[ \sigma (p), v(p)\left( s^{\sharp _G} \circ \sigma \right) + \rho _*\Big (\omega _{\sigma (p)}({\sigma _*}_p v(p))\Big )s^{\sharp _G}\left( \sigma (p)\right) \right] \\&= \bigg [\sigma (p), v(p)\left( s^{\sharp _G} \circ \sigma \right) + \bigg ({\Delta _{r,1}^{{\mathbb {C}}}}_*\left( (f_{\text {Spin}_{r,1}^0} \circ \sigma )^*\omega ^s_p (v(p))\right) \\&\qquad +{\rho _G}_*\Big ((f_G \circ \sigma )^*\omega ^G_p(v(p))\Big )\bigg )s^{\sharp _G}\left( \sigma (p)\right) \bigg ] \end{aligned}\nonumber \\ \end{aligned}$$

(2.16)

where $\sigma :U \rightarrow P_{\text {Spin}_{r,1}^0}+P_G$ is a local section of the direct product bundle and

$$\begin{aligned} \cdot ^{\sharp _G}:\Gamma (\wedge ^qT^*M\otimes S_G(M))\rightarrow \Omega ^q_\rho (P_{\text {Spin}_{r,1}^0} + P_G, V) \end{aligned}$$

denotes the isomorphism as of (2.2). Notice that as V is a ${\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$-module, even in this case we have a Clifford module multiplication

$$\begin{aligned} \cdot _{\Gamma , G}:{\mathfrak {X}}( M)\otimes \Gamma (S_G(M)) \rightarrow \Gamma (S_G(M)) \end{aligned}$$

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

(2.17)

with the analogous properties of the uncharged operators as after (2.13). Again, we can define the causal Green operator by posing .

Remark 2.4

Notice that the main difference between the local expressions (2.16) and (2.4) of the covariant derivatives, apart from the different spaces of section they are acting on, is given by the presence of the term

$$\begin{aligned} \left[ \sigma (p), {\rho _G}_*\left( (f_G \circ \sigma )^*\omega _p^G (v(p))\right) s^{\sharp _G}(\sigma (p))\right] \end{aligned}$$

which, when considering the Dirac operators ${{\textbf {D}}}^G$ and ${{\textbf {D}}}$, amounts to

$$\begin{aligned} ie^j(p) \cdot _{\Gamma ,G} \left[ {\sigma }(p), {\rho _G}_*\left( (f_G \circ {\sigma })^*\omega ^G_p(e_j(p))\right) s^{\sharp _G}({\sigma }(p))\right] \ . \end{aligned}$$

In the spirit of Remark 2.1, let us now exhibit an explicit expression in the simple case of Minkowski spacetime $({\mathbb {R}}^4, \eta )$. First of all, let us define the quantities

$$\begin{aligned} A_j(p) \overset{\cdot }{=}{\rho _G}_*\left( (f_G \circ {\sigma })^*\omega ^G_p(e_j(p))\right) \end{aligned}$$

which are maps $A_j :{\mathbb {R}}^4 \rightarrow \mathfrak {gl}({\mathbb {C}}^4)$. Then in the same way as before it follows that

$$\begin{aligned} {{\textbf {D}}}^G(s)(p) = i\gamma ^j(\partial _j s + A_j s) \ . \end{aligned}$$

$\square$

3 The Classical Møller Map on Field Configurations

3.1 The Entwining Maps Between Sections

In order to compare the uncharged theory and the charged one, 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

$$\begin{aligned} \begin{aligned} F&\overset{\cdot }{=}\left\{ f:P_{\text {Spin}_{r,1}^0} \rightarrow V, \ f \ \text {right}-\text {Spin}_{r,1}^0 \ \text {equivariant}\right\} \\ F_G&\overset{\cdot }{=}\left\{ f:P_{\text {Spin}_{r,1}^0} + P_G \rightarrow V, \ f \ \text {right}-\text {Spin}_{r,1}^0\times G \ \text {equivariant}\right\} \end{aligned} \end{aligned}$$

respectively. We recall that owing to [31, Sect. II] principal $\text {Spin}_{r,1}^0$-bundles over non-compact spin spacetimes admit a global section $\sigma :M \rightarrow P_{\text {Spin}_{r,1}^0}$, and are thus always trivial. To simplify the computations and the definition in this first case, we then make the following assumptions:

• (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. On $P_G$, we assume that a fixed Ehresmann connection $\omega _G \in \Omega ^1(P, {\mathfrak {g}})$ is given. As $P_G$ is trivial, there exists a global section $\sigma _G:M \rightarrow P_G$; we can use it to define the global section

  $$\begin{aligned} M\ni p\overset{{\tilde{\sigma }}}{\mapsto }(p, \sigma (p), \sigma _G(p)) \end{aligned}$$

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

  $$\begin{aligned} \begin{aligned} \varphi ^{s}:(M \times \text {Spin}_{r,1}^0) \times _{\Delta _{r,1}^{{\mathbb {C}}}} V&\rightarrow \quad M \times V\\ {[}(p,s), v]\qquad&\mapsto (p, \Delta _{r,1}^{{\mathbb {C}}}(s) v) \end{aligned} \end{aligned}$$

  and

  $$\begin{aligned} \begin{aligned} \varphi ^{s,G}:(M \times (\text {Spin}_{r,1}^0 \times G)) \times _{\rho } V&\rightarrow \quad M \times V\\ {[}(p, s,g), v] \qquad&\mapsto (p, \rho (s,g)v) \end{aligned}\ . \end{aligned}$$

  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 [24, Proposition 12.1.27] the Hermitian metric can be chosen to be such that Clifford multiplication satisfies

  $$\begin{aligned} \rho (\phi )^*= -\rho (\phi ^\dagger ), \qquad \phi ^\dagger = {\tilde{\alpha }}(\phi ^t) \end{aligned}$$

  where $\cdot ^{t}:{\mathscr {C}}\hspace{-1.5pt}\ell _{r,1} \rightarrow {\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$ denotes the transpose and ${\tilde{\alpha }}$ is the extension of ${\mathbb {R}}^{r,1}\ni {{\textbf {v}}}\rightarrow -{{\textbf {v}}}$ to ${\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$. We shall denote the Hermitian matrix associated with $(\cdot , \cdot )_V$ by $(h_{ij})$.

• (ii) As a second core assumption, we shall assume that the global gauge potential computed using the section ${\tilde{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}^0}+P_G$,

  $$\begin{aligned} (f_G\circ {\tilde{\sigma }})^*\omega ^G \in \Omega ^1(M, {\mathfrak {g}}) \end{aligned}$$

  has past-compact support. Clearly, this requirement depends on the gauge choice and is not gauge invariant; we shall expand on this in Remark 3.3.

• (iii) We will assume that the representation $\rho _G :G \rightarrow \text {GL}(V)$ commutes with the Clifford algebra representation inducing $\Delta _{r,1}^{{\mathbb {C}}}:\text {Spin}_{r,1}^0 \rightarrow \text {GL}(V)$. Notice that this greatly reduces the freedom in the group G: indeed, it can be shown [22, Theorem 4.3] that

  $$\begin{aligned} \begin{aligned} {\mathbb {C}}\hspace{-1.5pt}\ell _{r+1}\simeq M(2^{\lfloor (r+1)/2 \rfloor }, {\mathbb {C}})&\ \text {if} \ r+1 = 0 \mod 2 \\ {\mathbb {C}}\hspace{-1.5pt}\ell _{r+1}\simeq M(2^{\lfloor (r+1)/2 \rfloor }, {\mathbb {C}})\oplus M(2^{\lfloor (r+1)/2 \rfloor }, {\mathbb {C}})&\ \text {if} \ r+1 = 1 \mod 2 \qquad \end{aligned} \end{aligned}$$

  and that an irreducible ${\mathbb {C}}$-module for ${\mathscr {C}}\hspace{-1.5pt}\ell _{r,1}$ (which descends from an irreducible module for ${\mathbb {C}}\hspace{-1.5pt}\ell _{r+1}$) 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, $\rho _G(G)\subseteq {\mathbb {C}}\text {id}_V$. 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 $\sigma _G :M \rightarrow P_G$ allows us to construct the following maps:

Definition 3.1

Let us define the maps ${\mathcal {p}}:F_G \rightarrow F$,

$$\begin{aligned} F_G\ni f_g \mapsto ({\mathcal {p}}f_g)(p, s) \overset{\cdot }{=}f_g(p, s, \sigma _G(p)) \end{aligned}$$

(3.1)

and ${\mathcal {i}}:F \rightarrow F_G$,

$$\begin{aligned} F\ni f \mapsto ({\mathcal {i}}f)(p,s,g) \overset{\cdot }{=}\rho _G({\tilde{g}}^{-1})f(p,s) \end{aligned}$$

(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 $\text {right}-\text {Spin}_{r,1}^0$ equivariant, and given $f\in F$, ${\mathcal {i}}f$ is $\text {right}-\text {Spin}_{r,1}^0\times G$ equivariant: indeed,

$$\begin{aligned} \begin{aligned} \left( r_{{\overline{s}}}^*({\mathcal {p}}f_g)\right) (p,s)&= ({\mathcal {p}}f_g)(p, r_{{\overline{s}}}s) = f_g(p, r_{{\overline{s}}} s, \sigma _G(p)) = (r_{({\overline{s}}, \text {id}_G)}^*f_g)(p,s, \sigma _G(p)) \\&= \rho ({\overline{s}}^{-1}, \text {id}_G)f_g(p, s, \sigma _G(p)) = \Delta _{r,1}^{{\mathbb {C}}}({\overline{s}}^{-1})({\mathcal {p}}f_g)(p,s) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \left( r_{({\overline{s}},{\overline{g}})}^*({\mathcal {i}}f)\right) (p,s, g)&= ({\mathcal {i}}f)(p, r_{{\overline{s}}}s, r_{{\overline{g}}}g) = \rho _G({\overline{g}}^{-1})\rho _G({\tilde{g}}^{-1})f(p, r_{{\overline{s}}}s) \\&= \rho _G({\overline{g}}^{-1})\rho _G({\tilde{g}}^{-1})r_{{\overline{s}}}^*f(p,s) \\ {}&= \rho _G({\overline{g}}^{-1})\rho _G({\tilde{g}}^{-1}) \Delta _{r,1}^{{\mathbb {C}}}({\overline{s}}^{-1})f(p,s) \\&= \Delta _{r,1}^{{\mathbb {C}}}({\overline{s}}^{-1})\rho _G({\overline{g}}^{-1})\rho _G({\tilde{g}}^{-1})f(p,s) \\ {}&= \rho ({\overline{s}}^{-1}, {\overline{g}}^{-1})({\mathcal {i}}f)(p,s,g)\ . \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} {\mathcal {i}} :\Gamma (S(M))\rightarrow \Gamma (S_G(M))&\qquad {\mathcal {p}}:\Gamma (S_G(M))\rightarrow \Gamma (S(M))\\ s\hspace{0.5cm} \mapsto \hspace{0.3cm}({\mathcal {i}}s^\sharp )^{\flat _G}\hspace{0.3cm}&\qquad \hspace{1.1cm} s\hspace{0.5cm}\mapsto \hspace{0.3cm}({\mathcal {p}}s^{\sharp _G})^\flat \end{aligned} \end{aligned}$$

(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

$$\begin{aligned} \begin{aligned} \left\Vert s\right\Vert _{K,n}&\overset{\cdot }{=}\max _{0\le i\le n} {\sup _{p\in K}\left\Vert \left( {\nabla ^s}^i s\right) (p)\right\Vert _{S(M)\otimes T^*M^{\otimes i}}} \\ \left\Vert s\right\Vert _{K,n}^G&\overset{\cdot }{=}\max _{0\le i\le n} {\sup _{p\in K}\left\Vert \left( {\nabla ^{s,G}}^i s\right) (p)\right\Vert _{S_G(M)\otimes T^*M^{\otimes i}}}\ . \end{aligned} \end{aligned}$$

(3.4)

We shall indicate these topological spaces as and respectively.

Lemma 3.2

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

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 $\{(K_l, n_l)\}_{1\le l \le m}, \{(K_l^G, n_l^G)\}_{1\le l \le m^G}$ such that

$$\begin{aligned} \left\Vert {\mathcal {i}}s\right\Vert ^G_{K,n}\le k_{K,n} \max _{1\le l \le m}\left\Vert s\right\Vert _{K_l, n_l} \qquad \left\Vert {\mathcal {p}}s\right\Vert _{K,n}\le k_{K,n}^G \max _{1\le l \le m^G}\left\Vert s\right\Vert ^G_{K_l^G, n_l^G}\ . \end{aligned}$$

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

$$\begin{aligned} \left\Vert {\mathcal {i}}s\right\Vert ^G_{K,0}{} & {} = {\sup _{p\in K}}\left\Vert ({\mathcal {i}}s)(p)\right\Vert _{S_G(M)}\\{} & {} = \sup _{p\in K} \left\Vert \rho (\pi _{2}({\tilde{\sigma }}(p)))({\mathcal {i}}s)^{\sharp _G}({\tilde{\sigma }}(p))\right\Vert _{V} \\{} & {} = \sup _{p\in K} \left\Vert \rho _G(\pi _2\circ \sigma _G(p))\Delta _{r,1}^{{\mathbb {C}}}(\pi _2 \circ \sigma (p))s^\sharp (\sigma (p))\right\Vert _{V} \\{} & {} \le \sup _{p\in K}\left\Vert \rho _G\left( \pi _2 \circ {\sigma _G(p)}\right) \right\Vert _{\text {End}(V)} \sup _{p\in K}\left\Vert \Delta _{r,1}^{{\mathbb {C}}}(\pi _2 \circ \sigma (p))s^\sharp (\sigma (p))\right\Vert _V \\ {}{} & {} = k_{K, 0} \left\Vert s\right\Vert _{K, 0} \end{aligned}$$

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 $\{e_i\}_{1\le i\le \dim (M)}$ is a local pesudo-orthonormal basis for TM. In particular, using a section ${\widehat{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}^0} + P_G$ with ${\widehat{\sigma }} = r_g {\tilde{\sigma }}$,

$$\begin{aligned} \nabla _{e_i}^{s,G}({\mathcal {i}}s)(p){} & {} = \Big [{\widehat{\sigma }}(p), e_i(p)\Big (\rho _G\left( \pi _G \circ g(p)\right) ^{-1}s^\sharp (f_P \circ {\widehat{\sigma }}(p))\Big ) \\ {}{} & {} \quad + \Big ({\Delta _{r,1}^{{\mathbb {C}}}}_*\left( \left( (f_P \circ {\widehat{\sigma }})^*\omega ^s\right) _p\left( e_i (p)\right) \right) \\{} & {} \quad + {\rho _G}_*\left( \left( (f_G \circ {\widehat{\sigma }}) ^*\omega ^G\right) _p\left( e_i(p)\right) \right) ({\mathcal {i}}s)^{\sharp _G}({\widehat{\sigma }}(p))\Big ]\\{} & {} = \Big [{\widehat{\sigma }}(p), \rho _G\left( \pi _G \circ g(p)\right) ^{-1}\Big (e_i(p)\left( s^\sharp \circ f_P \circ {\widehat{\sigma }}\right) \\{} & {} \quad + {\Delta _{r,1}^{{\mathbb {C}}}}_*\left( \left( (f_P \circ {\widehat{\sigma }})^*\omega ^s\right) _p(e_i(p))\right) s^\sharp (\sigma (p))\Big )\Big ]\\{} & {} \quad + \Big [{\widehat{\sigma }}(p), e_i(p)\left( \rho _G\left( \pi _G \circ g(p)\right) ^{-1}\right) s^{\sharp }(f_P \circ {\widehat{\sigma }}(p))\\{} & {} \quad +{\rho _G}_*\left( \left( (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\right) (is)^{\sharp _G}({\widehat{\sigma }}(p))\Big ]\ . \end{aligned}$$

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

$$\begin{aligned} \left[ {\widehat{\sigma }}(p), {\rho _G}_*\left( \left( -(\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\right) ({\mathcal {i}}s)^{\sharp _G}({\widehat{\sigma }}(p))\right] \end{aligned}$$

(3.5)

where $\theta _G$ denots the Maurer–Cartan form of G. Using the notation $\varepsilon _i = g(e_i, e_i)$ we have

$$\begin{aligned} \left\Vert \nabla ^{s,G}({\mathcal {i}}s)(p)\right\Vert _{S_G(M)\otimes T^*M}^2{} & {} = \sum _{i=1}^{\dim M}\varepsilon _i\left\Vert \nabla ^{s,G}_{e_i}({\mathcal {i}}s)(p)\right\Vert _{S_G(M)}^2\\{} & {} \le k \sum _{i=1}^{\dim M}\varepsilon _i\left\Vert {\mathcal {i}}\circ \nabla ^s_{e_i} s (p)\right\Vert _{S_G(M)} ^2 \\{} & {} \quad +\, k\sum _{i=1}^{\dim M}\varepsilon _i\left\Vert \rho \left( \pi _2 \circ {\widehat{\sigma }}(p)\right) {\rho _G}_*\Big (\left( -(\pi _G \circ g)^*\theta _G\right\Vert \right. \\{} & {} \quad \left. +\, (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\Big ) ({\mathcal {i}}s)^{\sharp _G}(\sigma (p))_V^2 \ . \end{aligned}$$

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

$$\begin{aligned} k_1\sup _{p\in K}\left\Vert \nabla ^{s}s\right\Vert _{S(M)\otimes T^*M}^2 \le k_1 \left\Vert s\right\Vert _{K, 1}^2 \end{aligned}$$

while for the second term we have the upper bound

$$\begin{aligned} k_2\sup _{p\in K}\left\Vert s\right\Vert _{S(M)}^2 = k_2 \left\Vert s\right\Vert _{K,0}^2\le k_2 \left\Vert s\right\Vert _{K,1}^2\ . \end{aligned}$$

Therefore we can give the estimate

$$\begin{aligned} \left\Vert {\mathcal {i}}s\right\Vert ^G_{K,1} \le k_{K,1} \left\Vert s\right\Vert _{K,1}\ . \end{aligned}$$

We proceed by induction: suppose we have an inequality of the form $\left\Vert {\mathcal {i}}s\right\Vert ^G_{K,i}\le k_{K,i}\left\Vert s\right\Vert _{K,i}$ 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 $k_n\left\Vert s\right\Vert _{K,n}$, plus lower order terms in the covariant derivative of s and of the map ${\rho _G}_*\left( \left( -(\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\right)$ 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 ${\mathcal {p}}:\Gamma (S_G(M))\rightarrow \Gamma (S(M))$: indeed,

$$\begin{aligned} \left\Vert {\mathcal {p}}s\right\Vert _{K,0}{} & {} = \sup _{p\in K} \left\Vert ({\mathcal {p}}s)(p)\right\Vert _{S(M)}\\{} & {} = \sup _{p\in K}\left\Vert \Delta _{r,1}^{{\mathbb {C}}}(\pi _2 \circ \sigma (p))({\mathcal {p}}s)^\sharp \left( \sigma (p)\right) \right\Vert _V \\{} & {} = \sup _{p\in K}\left\Vert \Delta _{r,1}^{{\mathbb {C}}}(\pi _2 \circ \sigma (p))s^{\sharp _G}\left( {\tilde{\sigma }}(p)\right) \right\Vert _V \\{} & {} \le \sup _{p\in K} \left\Vert \rho _G(\pi _2 \circ \sigma _G(p))^{-1}\right\Vert _{\text {End}(V)}\\ {}{} & {} \quad \quad \sup _{p\in K}\left\Vert \rho _G(\pi _2 \circ \sigma _G(p))\Delta _{r,1}^{{\mathbb {C}}}(\pi _2 \circ \sigma (p))s^{\sharp _G}\left( {\tilde{\sigma }}(p)\right) \right\Vert _V \\{} & {} = k^G_{K,0}\sup _{p\in K} \left\Vert s\right\Vert _{S^G(M)} \end{aligned}$$

while for the case $n=1$ we have

$$\begin{aligned} \nabla ^s_{e_i}({\mathcal {p}}s)(p){} & {} = \left[ \sigma (p), e_i(p)\left( s^{\sharp _G}({\tilde{\sigma }}(p))\right) + {\Delta _{r,1}^{{\mathbb {C}}}}_*\Big ((\sigma ^*\omega ^s)_p (e_i(p))\Big )s^{\sharp _G}({\tilde{\sigma }}(p))\right] \\{} & {} = \Big [\sigma (p), e_i(p)\left( s^{\sharp _G}({\tilde{\sigma }}(p))\right) + {\Delta _{r,1}^{{\mathbb {C}}}}_*\Big ((\sigma ^*\omega ^s)_p (e_i(p))\Big )s^{\sharp _G}({\tilde{\sigma }}(p)) \\{} & {} \quad + {\rho _G}_*\Big ((\sigma _G^*\omega ^G)_p(e_i(p))\Big )s^{\sharp _G}({\tilde{\sigma }}(p))\\ {}{} & {} \quad - {\rho _G}_*\Big ((\sigma _G^*\omega ^G)_p(e_i(p))\Big )s^{\sharp _G}({\tilde{\sigma }}(p))\Big ]\ . \end{aligned}$$

Notice that

$$\begin{aligned} \begin{aligned} \Big [\sigma (p), e_i(p)(s^{\sharp _G} \circ {\tilde{\sigma }})&+ \Big ({\Delta _{r,1}^{{\mathbb {C}}}}_*\Big ((\sigma ^*\omega ^s)_p(e_i(p))\Big )\\&+{\rho _G}_*\Big ((\sigma _G^*\omega ^G)_p (e_i(p))\Big )\Big )s^{\sharp _G}({\tilde{\sigma }}(p))\Big ] = {\mathcal {p}}\left( \nabla _{e_i}^{s,G} s\right) (p)\ . \end{aligned} \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} \left\Vert \nabla ^s ({\mathcal {p}}s)(p)\right\Vert _{S(M)\otimes T^*M}^2&= \sum _{i=1}^{\dim M}\varepsilon _i \left\Vert \nabla ^s_{e_i} ({\mathcal {p}}s)(p)\right\Vert _{S(M)}^2 \\&\le k\sum _{i=1}^{\dim M} \varepsilon _i\left\Vert {\mathcal {p}}\left( \nabla _{e_i}^{s,G}s\right) (p)\right\Vert ^2_{S(M)}\\ {}&\quad +k\sum _{i=1}^{\dim M}\varepsilon _i\left\Vert \Delta _{r,1}^{{\mathbb {C}}}\left( \pi _2 \circ \sigma (p)\right) \right\Vert \\ {}&\quad {{\rho _G}_*\Big ((\sigma _G^*\omega ^G)_p(e_i(p)))\Big )s^{\sharp _G}({\tilde{\sigma }}(p))}_V^2\ . \end{aligned} \end{aligned}$$

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

$$\begin{aligned} k_1 \sup _{p\in K} \left\Vert \nabla ^{s,G}s\right\Vert _{S_G(M)\otimes T^*M} \le k_1 \left( {\left\Vert s\right\Vert ^G_{K,1}}\right) ^2 \end{aligned}$$

and the second term can be bounded from above by

$$\begin{aligned} k_2 \sup _{p\in K} \left\Vert s\right\Vert _{S_G(M)}^2 \le k_2 \left( {\left\Vert s\right\Vert ^G_{K,0}}\right) ^2\ . \end{aligned}$$

By reasoning as above by induction we have the desired result. $\square$

Remark 3.3

Notice how a key object in the proof of Lemma 3.2 is given by (3.5), which we recall here for a section $s\in \Gamma (S_G(M))$:

$$\begin{aligned} \left[ {\widehat{\sigma }}(p), {\rho _G}_*\left( \left( -(\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\right) s^{\sharp _G}({\widehat{\sigma }}(p))\right] \ . \end{aligned}$$

In the above expression, ${\widehat{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}^0} + P_G$ is a section which can be written as ${\widehat{\sigma }}=r_g {\tilde{\sigma }}$ in terms of the the global reference section and of a smooth function $g:M \rightarrow \text {Spin}_{r,1}^0 \times G$. The above expression is gauge independent: indeed, given another section $\overline{\sigma }:M \rightarrow P_{\text {Spin}_{r,1}^0} + P_G$ we have that $\overline{\sigma } = r_{g'} \circ {\widehat{\sigma }}$ where $g':M \rightarrow \text {Spin}_{r,1}^0 \times G$ and

$$\begin{aligned} f_P(r_{g'} \circ {\widehat{\sigma }}) = r_{\pi _{\text {Spin}}(g')}f_P({\widehat{\sigma }}) \qquad f_G(r_{g'} \circ {\widehat{\sigma }}) = r_{\pi _{G}(g')}f_G({\widehat{\sigma }})\ . \end{aligned}$$

The following relations holds between the different pullbacks of the connection form $\omega ^G$ and of the Maurer–Cartan form $\theta _G$:

$$\begin{aligned} \begin{aligned} (f_G \circ \overline{\sigma })^*\omega ^G&= (r_{\pi _G(g')} \circ f_G \circ {\widehat{\sigma }})^*\omega ^G = \text {Ad}_{\pi _G(g')^{-1}} \circ (f_G \circ {\widehat{\sigma }})^*\omega ^G + (\pi _G(g'))^*\theta _G \ , \\ (\pi _G(g g'))^*\theta _G&= (r_{\pi _G(g')} \circ \pi _G(g))^*\theta _G = \text {Ad}_{\pi _G(g')^{-1}}\circ (\pi _G(g))^*\theta _G + (\pi _G(g'))^*\theta _G\ . \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned}&\left[ \overline{\sigma }(p), {\rho _G}_*\Big (\left( -(\pi _G(gg'))^*\theta _G + (f_G \circ \overline{\sigma })^*\omega ^G\right) _p(e_i(p))\Big ) s^{\sharp _G}(\overline{\sigma }(p))\right] \\&=\Big [{\widehat{\sigma }}(p), \rho (g'(p)) \times {\rho _G}_*\Big (\text {Ad}_{\pi _G(g')^{-1}} \circ \left( (f_G \circ {\widehat{\sigma }})^*\omega ^G \right. \\ {}&\quad \left. - (\pi _G(g))^*\theta _G\right) \Big )_p(e_i(p))\rho ({g'}^{-1}(p))s^{\sharp _G}({\widehat{\sigma }}(p))\Big ] \\&= \left[ {\widehat{\sigma }}(p), {\rho _G}_*\Big (\left( -(\pi _G(g))^*\theta _G+ (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p)) \Big )s^{\sharp _G}({\widehat{\sigma }}(p))\right] \ . \end{aligned} \end{aligned}$$

Notice moreover that if the section ${\widehat{\sigma }}$ is such that ${\widehat{\sigma }}={\tilde{\sigma }}$ the above expression simplifies to

$$\begin{aligned} \left[ {\tilde{\sigma }}(p), {\rho _G}_*\Big (\left( \sigma _G^*\omega ^G\right) _p(e_i(p)) \Big )s^{\sharp _G}({\tilde{\sigma }}(p))\right] \end{aligned}$$

as the map $g:M \rightarrow \text {Spin}_{r,1}^0 \times G$ is constantly the identity. Clearly, whereas the above expression is past-compactly supported by the assumption (ii), the same doesn’t hold in general for

$$\begin{aligned} \left[ {\widehat{\sigma }}(p), {\rho _G}_*\left( \left( -(\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\right) ({\mathcal {i}}s)^{\sharp _G}({\widehat{\sigma }}(p))\right] \ . \end{aligned}$$

The compactness property holds for instance if $\pi _G \circ g:M \rightarrow G$ is constant outside of a past-compact region of M. As this property is fundamental in the construction of the (retarded) Møller map, we restrict ourselves to considering gauge potentials induced by sections ${\widehat{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}^0} + P_G$ which are such that $\pi _G \circ g$ is constant outside of a past-compact set, where $g:M \rightarrow \text {Spin}_{r,1}^0 \times G$ is such that ${\widehat{\sigma }} = r_g {\tilde{\sigma }}$. Having made this choice, we define the map to act pointwise as

$$\begin{aligned} (As)(p){} & {} \overset{\cdot }{=}ie^j(p) \cdot _{\Gamma ,G} \left[ {\widehat{\sigma }}(p), {\rho _G}_*\left( \left( -(\pi _G \circ g)^*\theta _G \right. \right. \right. \nonumber \\ {}{} & {} \quad \left. \left. \left. + (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) _p(e_i(p))\right) s^{\sharp _G}({\widehat{\sigma }}(p))\right] \ . \end{aligned}$$

(3.6)

We define

$$\begin{aligned} \text {supp}({\mathscr {A}},g)\overset{\cdot }{=}\text {supp}\left( -(\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G\right) \ . \end{aligned}$$

Notice that under the restriction on the possible gauge transformations, $\text {supp} ({\mathscr {A}},g)$ is always past-comapct. We now exhibit an important property of the map A defined in (3.6): to do so, we first show that commutes with the Clifford multiplication by sections of the cotangent bundle, that is, we show that

$$\begin{aligned} e^i \cdot _{\Gamma , G} {\mathcal {i}}(u) = {\mathcal {i}}\left( e^i \cdot _{\Gamma } u\right) \ . \end{aligned}$$

Consider the section ${\widehat{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}^0} + P_G$ inducing a section $f_P \circ {\widehat{\sigma }} :M \rightarrow P_{\text {Spin}_{r,1}^0}$; 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 $h:M \rightarrow \text {Spin}_{r,1}^0 \times G$ is such that ${\widehat{\sigma }} = r_h {\tilde{\sigma }}$. Then

$$\begin{aligned} \begin{aligned} e^i(p) \cdot _{\Gamma , G} {\mathcal {i}}(u)(p)&= \left[ {\widehat{\sigma }}(p), \left( g^{ij}(p) e_j(p)\right) \cdot _V \rho _G\left( (\pi _2 \circ h)(p)\right) ^{-1} u^\sharp (f_P\circ {\widehat{\sigma }}(p))\right] \\&= \left[ {\widehat{\sigma }}(p), \rho _G\left( (\pi _2 \circ h)(p)\right) ^{-1}\left( \left( g^{ij}(p) e_j(p)\right) \cdot _V u^\sharp (f_P\circ {\widehat{\sigma }}(p))\right) \right] \\&= {\mathcal {i}}(e^i \cdot _\Gamma u )(p) \ . \end{aligned} \end{aligned}$$

Notice that assumption (iii) was used in the second equality.

Thus, as shown in the proof of Lemma 3.2 we know that

$$\begin{aligned} {{\textbf {D}}}^G \circ {\mathcal {i}} = {\mathcal {i}} \circ {{\textbf {D}}}+ A \circ {\mathcal {i}}\ . \end{aligned}$$

(3.7)

$\square$

3.2 The Explicit Construction of the Classical Møller Map

We now use these maps, in particular to give an explicit formula for the retarded classical Møller operator, whose definition in the examined case we recall:

Definition 3.4

Let us consider an admissible gauge potential (as of Remark 3.3), i.e. an admissible section ${\widehat{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}^0}\times P_G$ such that ${\widehat{\sigma }}=r_g{\tilde{\sigma }}$. The retarded classical Møller map on field configurations is a map such that

$\mathrm{(i)}$:

${{\textbf {D}}}^G \circ R_A^- = {\mathcal {i}} \circ {{\textbf {D}}}$ ,

$\mathrm{(ii)}$:

$R_A^-(s)|_{M \setminus J^+(\text {supp}({\mathscr {A}},g))} = ({\mathcal {i}}s)|_{M \setminus J^+(\text {supp}({\mathscr {A}},g))}$ .

Remark 3.5

A similar construction can be made for the advanced Møller map , which is defined analogously to Definition 3.4 except that requirement (ii) now reads

$$\begin{aligned} R_A^+(s)|_{M \setminus J^-(\text {supp}({\mathscr {A}},g))} = ({\mathcal {i}}s)|_{M \setminus J^-(\text {supp}({\mathscr {A}},g))} \end{aligned}$$

In the following, we shall focus on the retarded classical Møller map which shall be denote simply with . The reader may refer to [32] for furhter details.

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

Theorem 3.6

The unique solution to the requirements in Definition 3.4 is given by

(3.8)

where is given by Eq. 3.6.

Proof

First of all, notice that the composition is well-defined thanks to the fact that $\text {supp}({\mathscr {A}},g)$ is past-compact: indeed, we know that

and as As vanishes outside of $\text {supp}({\mathscr {A}},g)$ for every we have that for every .

Thanks to the properties of the retarded propagator , we thus have that

and therefore the requirement (ii) in Definition 3.4 is fulfilled.

Using the property of the retarded propagator associated to ${{\textbf {D}}}^G$ we have ; therefore, by (3.7),

Therefore, also the requirement (i) in Definition 3.4 is fulfilled. Thus we can say that

As far as the uniqueness statement is concerned, we proceed as done in [33] and [34]: given , let us define ; then we have that $\psi$ satisfies the following:

$$\begin{aligned} {{\textbf {D}}}^G \psi = ({\mathcal {i}} \circ {{\textbf {D}}}) s \ ,\quad \psi |_{M \setminus J^+(\text {supp}({\mathscr {A}},g))} = ({\mathcal {i}}s)|_{M \setminus J^+(\text {supp}({\mathscr {A}},g))}\ . \end{aligned}$$

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 $\{0\}\times \Sigma \subseteq M \setminus J^+(\text {supp}({\mathscr {A}},g))$. Let $\{K_n\}_{n\in {\mathbb {N}}}$ be an invading sequence of compact sets for $\{0\}\times \Sigma$, and define

$$\begin{aligned} {\widehat{K}}_n \overset{\cdot }{=}D(K_n) \cap [-n, n] \times \Sigma \end{aligned}$$

where $D(K_n)$ denotes the Cauchy development of $K_n$. We then consider the family , with $\chi _n \equiv 1$ on ${\widehat{K}}_n$. Using these family, we consider

Notice that ${{\textbf {D}}}^G \circ {{\textbf {D}}}^G$ is normally hyperbolic, and thus the above system admits a unique solution which depends continuously on the initial data [25]. Notice in particular that on $M \setminus J^+(\text {supp}({\mathscr {A}},g))$ the map

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

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

$$\begin{aligned} \phi (p)\overset{\cdot }{=}\phi _n(p) \ \text {with} \ n \ \text {such that} \ p\in {\widehat{K}}_n \end{aligned}$$

and $\psi \overset{\cdot }{=}{{\textbf {D}}}^G \phi$ we have that ${{\textbf {D}}}^G \psi = ({{\textbf {D}}}^G \circ {{\textbf {D}}}^G) \phi = ({\mathcal {i}} \circ {{\textbf {D}}})s$ and

$$\begin{aligned} \psi |_{M \setminus J^+(\text {supp}({\mathscr {A}},g))} = ({{\textbf {D}}}^G \circ \phi )|_{M \setminus J^+ (\text {supp}({\mathscr {A}},g))} = ({\mathcal {i}}s)|_{M \setminus J^+ (\text {supp}({\mathscr {A}},g))}\ . \end{aligned}$$

By the uniqueness, the $\psi$ above is unique for every s, and therefore $R_A$ is as well. $\square$

Remark 3.7

Notice that admits both a right and left inverse. Let us consider the map

(3.9)

and consider

It is easy to see that ${\mathcal {i}}\circ {\mathcal {p}} = \text {id}$ and ${\mathcal {p}} \circ {\mathcal {i}} = \text {id}$ on and respectively, while by the previous proof we know that $A \circ {\mathcal {i}} = {{\textbf {D}}}^G \circ {\mathcal {i}} - {\mathcal {i}} \circ {{\textbf {D}}}$; combining these observations we conclude that

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 we then infer that

In the same way, using the fact that

$$\begin{aligned} ({\mathcal {p}} \circ A)(s){} & {} = ie^j(p)\cdot _{\Gamma ,G} \left[ {\sigma }(p), {\rho _G}_*\Big (\left( \sigma _G^*\omega ^G\right) _p(e_j(p)) \Big )s^{\sharp _G}({\tilde{\sigma }}(p))\right] \\ {}{} & {} = {\mathcal {p}}\circ {{\textbf {D}}}^G - {{\textbf {D}}}\circ {\mathcal {p}} \end{aligned}$$

one can show that

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 Sect. 3.1 and the classical Møller map on field configurations $R_A$, whose explicit expression is given in Theorem 3.6, 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,

$$\begin{aligned} \begin{aligned} \left( ({\mathcal {i}}t), s\right) _G (p)&= \left( \rho (\pi _2 \circ {\widehat{\sigma }}(p))({\mathcal {i}}t)^{\sharp _G}({\widehat{\sigma }}(p)),\rho (\pi _2 \circ {\widehat{\sigma }}(p))s^{\sharp _G}({\widehat{\sigma }}(p)) \right) _V \\&= \left( \Delta _{r,1}^{{\mathbb {C}}}\left( f_P\circ {\widehat{\sigma }}(p) \right) \rho _G\left( f_G\circ {\widehat{\sigma }}(p)\right) ({\mathcal {i}}t)^{\sharp _G}( {\widehat{\sigma }}(p)), \right. \\&\quad \times \left. \Delta _{r,1}^{{\mathbb {C}}}\left( f_P\circ {\widehat{\sigma }}(p)\right) \rho _G(f_G \circ {\widehat{\sigma }}(p))s^{\sharp _G}\left( {\widehat{\sigma }}(p)\right) \right) _V \\&= \left( \Delta _{r,1}^{{\mathbb {C}}}\left( f_P\circ {\widehat{\sigma }}(p) \right) \rho _G(g(p))^{-1}t^{\sharp }( {\sigma }(p)), \right. \\&\quad \times \left. \Delta _{r,1}^{{\mathbb {C}}}\left( f_P\circ {\widehat{\sigma }}(p)\right) \rho _G(g(p))^{-1}s^{\sharp _G}\left( {\tilde{\sigma }}(p)\right) \right) _V\\&= \left( \Delta _{r,1}^{{\mathbb {C}}}\left( \pi _2 \circ \sigma (p) \right) t^\sharp ({\sigma }(p)), \Delta _{r,1}^{{\mathbb {C}}}\left( \pi _2 \circ \sigma (p)\right) s^{\sharp _G}\left( {\tilde{\sigma }}(p)\right) \right) _V \\&= \left( \Delta _{r,1}^{{\mathbb {C}}}\left( \pi _2 \circ \sigma (p) \right) t^\sharp ({\sigma }(p)),\Delta _{r,1}^{{\mathbb {C}}}\left( \pi _2 \circ \sigma (p)\right) ({\mathcal {p}}s)^{\sharp }\left( {\sigma }(p)\right) \right) _V \\&= \left( t, ({\mathcal {p}}s), \right) (p) \end{aligned} \end{aligned}$$

where we have used the fact that $G = U(1)$ and the G-equivariance of the sections.

We now move on to the behaviour of the map 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, if we define the quantity

$$\begin{aligned} A_k(p)\overset{\cdot }{=}{\rho _G}_*\left( \left( (-\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G_p\right) (e_k(p))\right) \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned} \left( (At), s\right) _G (p)&= \left( (At)(p), s(p)\right) _G \\&= \left( i(g^{kj}(p)e_{j}(p)) \cdot _V A_k(p)t^{\sharp _G}({\widehat{\sigma }}(p)), s^{\sharp _G}({\widehat{\sigma }}(p))\right) _V \\&= -\left( A_k(p)t^{\sharp _G}({\widehat{\sigma }}(p)) ,i(g^{kj}(p)e_{j}(p)) \cdot _V s^{\sharp _G}({\widehat{\sigma }}(p))\right) _V \\&= \left( t^{\sharp _G}({\widehat{\sigma }}(p)),iA_k(p)\left( (g^{kj}(p)e_{j}(p)) \cdot _V s^{\sharp _G}({\widehat{\sigma }}(p))\right) \right) _V \\&= \left( t^{\sharp _G}({\widehat{\sigma }}), i(g^{kj}(p)e_{j}(p)) \cdot _V \left( A_k(p) s^{\sharp _G}({\widehat{\sigma }})\right) \right) _V \\&= \left( t(p), (As)(p)\right) _G = \left( (As), t\right) _G (p)\ . \end{aligned} \end{aligned}$$

Let us then consider the formal adjoint of the classical Møller map on field configurations, which has the following behaviour:

Notice that as is linear, using the antilinear isomorphisms between a vector bundle and its conjugate bundle

$$\begin{aligned} C:\overline{S(M)} \rightarrow {S(M)} \qquad C_G:\overline{S_G(M)}\rightarrow S_G(M) \end{aligned}$$

we can naturally induce a linear map , ${\overline{R}}_A \overset{\cdot }{=}C_G^{-1}\circ R_A \circ C$, whose formal adjoint is given by . In particular, it holds that ${{\overline{R}}_A}^*= \overline{R_A^*}$. In light of the previous equalities, we can write

Notice that the antilinear isomorphisms can be used to define

These maps, as well as the map satisfy the same results as the ones previously proven.

Proposition 4.1

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

(4.1)

where the composition of maps appearing on the right-hand sides are restricted to .

Proof

Let us denote with the operators on the right-hand sides of the equalities in (4.1). It is easy to see that on it holds that

and

The same relations for can be obtained in the following way: we know that ${{{\textbf {D}}}^G}^*= C_G^{-1}\circ {{\textbf {D}}}^G \circ C_G$ and that ; therefore given any ,

As the above equality holds for any functions , , we infer that . The other equality can be obtained in a similar fashion.

To conclude the proof, it suffices to show that

We proceed for the retarded propagator , as the proof for the advanced one is entirely analogous. In particular, we shall prove that

To this end, it suffices to show that ; therefore, let , and let us consider two Cauchy surfaces $\Sigma _1, \Sigma _2 \subseteq M$ such that

$\mathrm{(i)}$:

$\Sigma _2\subseteq J^+(\Sigma _1)$;

$\mathrm{(ii)}$:

$\Sigma _1 \cap \Sigma _2 = \varnothing$;

$\mathrm{(iii)}$:

$\left( \text {supp}({\mathscr {A}}) \cup \{p\} \cup {\text {supp} }(u)\right) \cap J^+(\Sigma _1) = \varnothing$.

We then consider a smooth function such that

$$\begin{aligned} \varphi \equiv 1 \ \text {on} \ J^-(\Sigma _1), \qquad \varphi \equiv 0 \ \text {on} \ J^+(\Sigma _2) \end{aligned}$$

which we use to define the maps

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 is compact. Then

Now, notice that when evaluated at p, the above expression is zero: indeed, we know that , and due to the properties of we also know that ${{\textbf {D}}}(\varphi ) \equiv 0$ on $J^-(\Sigma _1)$ and $J^+(\Sigma _2)$, and therefore $({\mathcal {i}} \circ {{\textbf {D}}}(\varphi ) \circ S_- \circ {\mathcal {p}})(u)$ is supported in $J^+(\Sigma _1)\cap J^-(\Sigma _2)$. But then 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

which is equal to zero when evaluated at p, and

which vanishes as $\text {supp}({\mathscr {A}}) \cap {\text {supp} }(1-\varphi ) = \varnothing$. Thus , and we conclude. $\square$

Corollary 4.2

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

(4.2)

Proof

Let us define ; then

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

$$\begin{aligned} {{\overline{R}}_A}^*-{\mathcal {p}} = {{\overline{R}}_A}^*\circ o^\dagger \circ {\mathcal {p}} \qquad {{\overline{R}}_A}^*-{\mathcal {p}} = {\mathcal {p}} \circ o^\dagger \circ {{\overline{R}}_A}^*\end{aligned}$$

where . Now, using these equalities we have on

Proposition 4.1 then entails that on we have that

$\square$

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 ${{\textbf {D}}}$ and ${{\textbf {D}}}^G$, we now examine the behaviour of Hadamard bidistributions when coupled with the Møller maps. This is of particular signficance because of the twofold importance of Hadamard bidistribution in the context of perturbative algebraic quantum field theory: indeed, on one hand these are the 2-point functions of the analogous of vacuum states in curved spacetimes, and on the other (as shall be shown in Sect. 5.2) Hadamard bidistributions are the right tools to construct topological $*$-algebras of observables containing meaningful interactions and to construct Wick powers (see for instance [35, 36])

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 and ; the same holds for sections of $S^{\oplus }_G(M)$.

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

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

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

Using the Dirac operator and its adjoint ${{\textbf {D}}}^*= C^{-1} \circ {{\textbf {D}}}\circ C$ as well as the causal propagators and we construct the operators

Notice that , which is the causal propagator for ${{\textbf {D}}}^\oplus$, is formally self-adjoint: indeed, given we have

Therefore, the distribution given by the Schwartz kernel theorem,

is symmetric. Analogous extensions can be made in the case of the charged spinor bundle $S_G(M)$.

The Møller map as well as its conjugate , which satisfy Definition 3.4 and Theorem 3.6 (with suitable modifications), can be combined into one Møller map ,

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

$$\begin{aligned} {{\mathcal {R}}_A}^*= {\overline{R}}_A^*\oplus R_A^*\ . \end{aligned}$$

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

Definition 4.3

A distribution satisfies the Hadamard two-point condition if given any two sections we have:

$\mathrm{(i)}$:

$\zeta (({{\textbf {D}}}^\oplus u), {v}) = 0$;

$\mathrm{(ii)}$:

;

$\mathrm{(iii)}$:

we require that

$$\begin{aligned} \begin{aligned} \text {WF}(\zeta ) =&\Big \{(x, y, \xi _x, -\xi _y)\in T^*M^2 \setminus z(M^2) \ | \ (x,\xi _x) \sim (y, \xi _y)\\&\qquad \text {or} \ x=y, \xi _x=\xi _y, \ \xi _x \triangleright 0\Big \} \end{aligned} \end{aligned}$$

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 [8] and [12].

Proposition 4.4

If is a distribution satisfying the Hadamard two-point condition, then

$$\begin{aligned} \zeta _G(\cdot , \cdot )\overset{\cdot }{=}\zeta ({{\mathcal {R}}_A}^*\cdot , {{\mathcal {R}}_A}^*\cdot ) \end{aligned}$$

(4.3)

is a distribution in satisfying the Hadamard two-point condition.

Proof

First of all, notice that $\zeta _G$ defined as in (4.3) is well-defined: indeed, ${\overline{R}}_A^*$ and ${R_A}^*$ are continuous with respect to the inductive limit topology of and , being linear maps which are sequentially continuous. Moreover, they map compactly supported smooth functions to compactly supported smooth functions: indeed, if we consider the expression of ${\overline{R}}_A^*$ when acting on we have

Recall ; therefore we have that

where $\Sigma$ is a Cauchy surface such that $\text {supp}({\mathscr {A}},g)\subseteq J^+(\Sigma )$, the support being past-compact. But it holds [26, Corollary A.5.4] that

$$\begin{aligned} J^-(\text {supp}(u))\cap J^+(\Sigma ) \subset \subset M \end{aligned}$$

and so is compact. This result directly translates to the map ${{\mathcal {R}}_A}^*$. Let us now prove that the three requirements are satisfied:

(i):

it is easy to see that $\zeta _G({{{\textbf {D}}}^G}^\oplus u, {v})=0$; indeed, by Definition 3.4 and Theorem 3.6 we have

$$\begin{aligned} R_A^*\circ {{{\textbf {D}}}^{G}}^*=({{\textbf {D}}}^G \circ R_A)^*= ({\mathcal {i}} \circ {{\textbf {D}}})^*= {{\textbf {D}}}^*\circ {\mathcal {i}}^*\end{aligned}$$

and therefore ${{\mathcal {R}}_A}^*\circ {{{\textbf {D}}}^G}^\oplus = ({{\textbf {D}}}\circ {\mathcal {p}})\oplus (-{{\textbf {D}}}^*\circ {\mathcal {i}}^*)$; this entails that

$$\begin{aligned} \zeta _G({{{\textbf {D}}}^G}^\oplus u, {v}) = \zeta \left( {{{\mathcal {R}}_A}}^*( {{{\textbf {D}}}^G}^\oplus u), {{\mathcal {R}}_A}^*{v}\right) = \zeta ({{\textbf {D}}}^\oplus (({\mathcal {p}} \oplus {\mathcal {i}}^*) u), {{\mathcal {R}}_A}^*{ v}) = 0\ . \end{aligned}$$

(ii):

We know that $\zeta _G(u,{v}) = \zeta ({{\mathcal {R}}_A}^*u, {{\mathcal {R}}_A}^*{v})$, and by using Corollary 4.2 we have

(iii):

We know that $\zeta _G$ is a bisolution of ${{{\textbf {D}}}^G}^\oplus$, and that the principal symbol of ${{{\textbf {D}}}^G}^\oplus$ coincides with that of ${{\textbf {D}}}^\oplus$; therefore by [37, Theorem 6.1.1] we know that the wavefront set of $\zeta _G$ coincides with that of $\zeta$, being determined by the Hamiltonian flow associated to the principal symbol of ${{{\textbf {D}}}^G}^\oplus$.

$\square$

5 The Classical Møller Map on the Observable Algebras

5.1 The Poisson $*$-Algebras of Observables

Our goal is to pass the classical Møller map on field configurations $R_A$, presented in Definition 3.4 and Theorem 3.6, to the algebras of observables of the charged and uncharged Dirac field. We briefly recall the construction of said topological algebras, as presented in [7, 8, 38] and in Sect. 2 of [13].

Let $E\overset{\pi }{\rightarrow }\ M$ 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 , that is, the graded algebra

We can consider the spaces of homogeneous elements 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

as well as the configuration space

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

$$\begin{aligned} (u_1 \wedge \cdots \wedge u_p)^*\overset{\cdot }{=}u_p^*\wedge \cdots \wedge u_1^*\end{aligned}$$

on homogeneous elements in , by extending the above by continuity to and by linearity to ${\mathcal {C}}(E)$.

We are interested in antisymmetric functionals on the space of sections ; these can be interpreted as a sequence $\{F_p\}_{p\in {\mathbb {N}}}$ of linear and continuous functionals on , that is, a sequence of elements such that

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

Thus we define the space of fermionic functionals as

$$\begin{aligned} {\mathcal {F}}(E) \overset{\cdot }{=}\prod _{p\in {\mathbb {N}}} {\mathcal {F}}^p(E)\ . \end{aligned}$$

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

$$\begin{aligned} {\langle {F, u}\rangle } \overset{\cdot }{=}\sum _{p\in {\mathbb {N}}} {\langle {F_p, u_p}\rangle } \ \text {for all} \ F \in {\mathcal {F}}(E), \ u\in {\mathcal {C}}(E)\ . \end{aligned}$$

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 $\{p_u\}_{u\in {\mathcal {C}}(E)}, \ p_u(F) = \left|F(u)\right|$, 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 by

$$\begin{aligned} \begin{aligned} (F \wedge G)_p (u_1 \wedge \cdots \wedge u_p)&\overset{\cdot }{=}\sum _{\sigma \in S_p} \text {sgn}(\sigma ) \sum _{k=0}^p \frac{1}{k! (p-k)!} F_k(u_{\sigma (1)}\wedge \cdots \wedge u_{\sigma (k)})\\ {}&\qquad G_{p-k}(u_{\sigma (k+1)}\wedge \cdots \wedge u_{\sigma (p)})\ . \end{aligned} \end{aligned}$$

The object above is then extended by linearity and continuity to elements in , 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 ${\cdot }^{*}:{\mathcal {F}}(E)\rightarrow {\mathcal {F}}(E)$,

$$\begin{aligned} \{F_p\}_{p\in {\mathbb {N}}}\mapsto \left\{ F_p^*\right\} _{p\in {\mathbb {N}}}, \qquad (F_p^*)(u_p) \overset{\cdot }{=}\overline{F_p(u_p^*)}\ . \end{aligned}$$

(5.1)

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

$$\begin{aligned} d_h F_p(u) \overset{\cdot }{=}F_p(h\wedge u) \end{aligned}$$

and is then extended by continuity to , thus yielding a linear and continuous map , i.e. $d_h F_p\in {\mathcal {F}}^{p-1}(E)$. One can then extend the map $d_h :{\mathcal {F}}^{p}(E)\rightarrow {\mathcal {F}}^{p-1}(E)$ to the whole algebra ${\mathcal {F}}(E)$ by considering

$$\begin{aligned} d_h :F = \{F_p\}_{p\in {\mathbb {N}}} \mapsto d_h F \overset{\cdot }{=}\{d_h F_p\}_{p\in {\mathbb {N}}}\ . \end{aligned}$$

It is easy to see that for any , $d_h$ is a graded derivation. One can also consider higher order derivatives by iterating the left derivative: given $F_p\in {\mathcal {F}}^p(E)$, $k\le p$ and , we define for

$$\begin{aligned} d^{k}_{h_1, \ldots , h_k} F_p(u) = F_p(h_k \wedge \cdots \wedge h_1 \wedge u) \end{aligned}$$

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

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

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 ${E^*}^{\boxtimes k}\rightarrow M^k$, that is, an object of , where ${\widehat{\otimes }}_\pi$ denotes the completion of the tensor product in the projective topology¹.

In order to proceed with the quantization, one needs to restrict the $*$-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

$$\begin{aligned} \text {WF}(F^{(n)}_{u})\subseteq \Xi _n \overset{\cdot }{=}T^*\overset{\cdot }{M^n}\ \setminus \bigcup _{(p_1, \ldots , p_n)\in M^n} \left( \overline{V_{p_1}^+} \times \cdots \times \overline{V_{p_n}^+}\right) \cup \left( \overline{V_{p_1}^-} \times \cdots \times \overline{V_{p_n}^-}\right) \end{aligned}$$

for every $u\in {\mathcal {C}}(E)$, where with $\overline{V_p^\pm }$ we denote the closure of the future/past causal cone in $T_p^*M$. We endow this space with the initial locally convex topology induced by the family of linear maps $\{\ell _{k, u}\}$,

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

with $\{\Gamma _{k,n}\}_{n\in {\mathbb {N}}}$ a sequence of closed cones in $T^*M^k$ such that $\Gamma _{k,n} \subset \overset{\circ }{\Gamma }_{k,n+1}$ and $\cup _n \Gamma _{k,n} = \Xi _k$, and where is endowed with the usual Hörmander topology. Using the causal propagator 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 ,

(5.2)

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

Moreover, notice that this object is actually a smooth function: indeed, using [17, Theorem 8.2.13] one obtains that

Therefore, we can consider it as an object in , 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

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

and then extended to the whole of by continuity. This procedure is then extended to non-homogeneous functionals by considering ,

(5.3)

for . Then the Peierls’ bracket is given by ; notice that on homogeneous elements $F\in {\mathcal {A}}^p(E)$, $G\in {\mathcal {A}}^q(E)$ we have the desired graded anticommutativity

as well as the graded Jacobi identity.

5.2 Deformation Quantization and the Classical Møller Maps

Starting from the Poisson algebra , the quantization proceeds in the following way: we consider the $*$-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 $F\in {\mathcal {A}}^p(E)$, $G\in {\mathcal {A}}^q(E)$

Given a Hadamard bidistribution (see Definition 4.3) and two functionals $F, G\in {\mathcal {A}}(E)$, we consider the fermionic functional

$$\begin{aligned} \begin{aligned} \Gamma _\zeta ^n(G,F)&\overset{\cdot }{=}\left( \frac{i}{2}\right) ^n G^{(n)} \wedge \left( \zeta ^{\boxtimes n} *F^{(n)}\right) \ \forall \ n\in {\mathbb {N}}\ . \end{aligned} \end{aligned}$$

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

$$\begin{aligned} G \star F \overset{\cdot }{=}\sum _{n\in {\mathbb {N}}} \hbar ^n \Gamma _\zeta ^n (G,F)\in {\mathcal {A}}(E)[\![\hbar ]\!]\ . \end{aligned}$$

(5.4)

This product can then be easily extended to the whole topological $*$-algebra of formal power series. Now, let ${\mathcal {A}}(S)[\![\hbar ]\!]$ and ${\mathcal {A}}(S_G)[\![\hbar ]\!]$ be the topological $*$-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 ${\mathscr {R}}_A :{\mathcal {A}}(S_G)[\![\hbar ]\!]\rightarrow {\mathcal {A}}(S)[\![\hbar ]\!]$ by considering the pullbacks induced by , where ${\mathcal {R}}_A^{\wedge p}$ is the continuous extension of the map

$$\begin{aligned} {\mathcal {R}}_A^{\wedge p} (u_1 \wedge \cdots \wedge u_p) \overset{\cdot }{=}{\mathcal {R}}_A(u_1) \wedge \cdots \wedge {\mathcal {R}}_A (u_p) \end{aligned}$$

to . Now,

Theorem 5.1

${\mathscr {R}}_A$ is a well-defined $*$-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

$$\begin{aligned} \text {WF}\left( ({\mathscr {R}}_A(F))_u^{(n)}\right) \subseteq \Xi _n \ \text {for \ all}\ u\in {\mathcal {C}}(S^\oplus (M))\ \text {and} \ n\in {\mathbb {N}}. \end{aligned}$$

(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 . This is defined as the wavefront set of the distribution $r_A$, where is obtained thanks to Schwartz’s kernel theorem and satisfies

To compute its wavefront set, we proceed locally, as presented in [17]. Namely, let $\{e_i\}_{1\le i \le \text {rank}(S_G(M))}$ be a local² frame for $S_G(M)$ and $\{f_j\}_{1\le j \le \text {rank}(S(M))}$ a local frame for S(M); a local frame for $S^\oplus (M)$ and $S_G^\oplus (M)$ is then given by

$$\begin{aligned} \{f_1, \ldots , f_{\text {rank}(S(M))}, {\overline{f}}_1, \ldots , {\overline{f}}_{\text {rank}(S(M))}\} \quad \{e_1, \ldots , e_{\text {rank}(S_G(M))}, {\overline{e}}_1, \ldots , {\overline{e}}_{\text {rank}(S_G(M))}\}\nonumber \\ \end{aligned}$$

(5.6)

respectively. Using these, we can write locally

$$\begin{aligned} r_A = {r_A}_{ji} {f}^j \boxtimes e^i \end{aligned}$$

with and where $\{e^i\}_{i}$ and $\{f^j\}_{j}$ denote the dual frames of $S_G^{\oplus }(M)^*\simeq \overline{S_G(M)}\oplus S_G(M)$ and $S^\oplus (M)^*\simeq \overline{S(M)}\oplus S(M)$ with respect to those in (5.6). Then

$$\begin{aligned} \text {WF}({\mathcal {R}}_A) \doteq \text {WF}(r_A)=\bigcup _{\begin{array}{c} 1\le i \le 2\text {rank}(S_G(M)) \\ 1\le j \le 2\text {rank}(S(M)) \end{array}} \text {WF}\left( {r_A}_{ji}\right) \end{aligned}$$

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

$$\begin{aligned} {{\widehat{\phi }}_\alpha }^{-1}(\text {WF}(\phi _\alpha ^*{r_A}_{ji})) \end{aligned}$$

with ${\widehat{\phi }}_\alpha :T^*M_{U_\alpha }^2\rightarrow U_\alpha ^2 \times {\mathbb {R}}^{\text {dim}(M^2)}$. Moreover, due to the properties of the wavefront set, we can study $\text {WF}({\mathcal {R}}_A)$ by studying separately the two terms Let us thus first consider the map ; then we have that

$$\begin{aligned} \begin{aligned} \int _M \left( {\mathcal {i}}^\oplus u, v\right) _G\, d\mu _g&= \int _M \left( {\mathcal {i}}u_1, v_2\right) _G + \left( v_1, \overline{{\mathcal {i}}}u_2\right) _G\, d\mu _g \\&= \int _M \left( u_1^\sharp (\sigma (p)), v_2^{\sharp _G}({\tilde{\sigma }}(p))\right) _V\, d\mu _g \\&\quad +\int _M \left( v_1^{\sharp _G}({\tilde{\sigma }}(p)), u_2^{\sharp }(\sigma (p))\right) _V\, d\mu _g \\&= \int _M {u_1^\sharp }^j(\sigma (p))h_{ji}{v_2^{\sharp _G}}^i({\tilde{\sigma }}(p))\, d\mu _g \\ {}&\quad + \int _M {v_1^{\sharp _G}}^i({\tilde{\sigma }}(p))h_{ij} {u_2^\sharp }^j(\sigma (p))\, d\mu _g \end{aligned} \end{aligned}$$

that is,

$$\begin{aligned} i_{ji}={\left\{ \begin{array}{ll} \delta _\Delta h_{ji}\, &{} j \le \text {rank}(S(M)), i \le \text {rank}(S_G(M)) \\ \delta _\Delta {\overline{h}}_{(j-\text {rank}(S(M)))(i-\text {rank}(S_G(M))}\, &{} j> \text {rank}(S(M)), i > \text {rank}(S_G(M)) \end{array}\right. }\ . \end{aligned}$$

Therefore,

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

As far as the map is concerned, we just need to compute the wavefront set of $A^\oplus$. In particular, by considering a gauge choice ${\widehat{\sigma }}:M \rightarrow P_{\text {Spin}_{r,1}+P_G}$ and defining

$$\begin{aligned} A_k(p)\overset{\cdot }{=}{\rho _G}_*\left( \left( (-\pi _G \circ g)^*\theta _G + (f_G \circ {\widehat{\sigma }})^*\omega ^G_p\right) (e_k(p))\right) \end{aligned}$$

we can proceed as done in the case of ${\mathcal {i}}^\oplus$:

$$\begin{aligned} \begin{aligned} \int _M (A^\oplus u, v)_G\, d\mu _g&=\int _M \left( i(g^{kj}(p)e_{j}(p)) \cdot _V A_k(p)u_1^{\sharp _G}({\widehat{\sigma }}(p)),v_2^{\sharp _G}({\widehat{\sigma }}(p))\right) _V\, d\mu _g \\&\quad + \int _M \left( v_1^{\sharp _G}({\widehat{\sigma }}(p)),\overline{i(g^{kj}(p)e_{j}(p))} \cdot _V \left( \overline{A_k(p)}u_2^{\sharp _G}({\widehat{\sigma }}(p))\right) \right) _V\, d\mu _g \\&= \int _M \left( i(g^{kj}(p)e_{j}(p)) \cdot _V A_k(p)\right) ^l_j {u_1^{\sharp }}^j({\widehat{\sigma }}(p)) h_{li}{v_2^{\sharp _G}}^i({\widehat{\sigma }}(p))\, d\mu _g \\&\quad + \int _M \left( \overline{i(g^{kj}(p)e_{j}(p)) \cdot _V A_k(p)}\right) ^l_i {v_1^{\sharp _G}}^i({\widehat{\sigma }}(p)){u_2^{\sharp _G}}^j({\widehat{\sigma }}(p))h_{lj}\ . \end{aligned} \end{aligned}$$

Thus $A_{ji}=\delta _\Delta f_{ji}$, $f_{ji}$ given by

$$\begin{aligned} f_{ji} = {\left\{ \begin{array}{ll}\left( i(g^{kj}(p)e_{j}(p)) \cdot _V A_k(p)\right) ^l_j h_{li}\, &{} j, i \le d \\ \left( \overline{i(g^{ij}(p)e_{j}(p)) \cdot _V A_k(p)}\right) ^l_{i-d} {\overline{h}}_{(j-d)l}\, &{} j, i > d\ , \end{array}\right. } \end{aligned}$$

with $d=\text {rank}(S_G(M))$. Using the fact that $(f_G \circ {\widehat{\sigma }})^*\omega ^G$ is assumed to be supported in $\text {supp}({\mathscr {A}},g)$, we then have that

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

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 ${\text {supp} }({\mathcal {i}}^\oplus ) \subseteq \Delta$, we have that ${\text {supp} }({\mathcal {i}}^\oplus ) \ni (x,y)\mapsto y$ is proper, and moreover

$$\begin{aligned} \begin{aligned} \text {WF}'({\mathcal {i}}^\oplus )_M&= \left\{ (y, \xi _y) \ | \ (x,y,0, -\xi _y)\in \text {WF}({\mathcal {i}}^\oplus )\right\} = \varnothing \\ \text {WF}(A^\oplus )_M&= \left\{ (x, \xi _x) \ | \ (x,y,\xi _x, 0) \in \text {WF}(A^\oplus )\right\} = \varnothing \ . \end{aligned} \end{aligned}$$

Thus $\text {WF}'({\mathcal {i}}^\oplus )_M \cap \text {WF}(A^\oplus )_M = \varnothing$, and [17, Theorem 8.2.14] gives us the well-posedness of $A^\oplus \circ {\mathcal {i}}^\oplus$ as well as

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

Let us now consider ; thanks to [39] we know that

As evidently $\text {WF}'(A^\oplus \circ {\mathcal {i}}^\oplus )_M = \varnothing$ and ${\text {supp} }(A^\oplus \circ {\mathcal {i}}^\oplus )\ni (x,y) \mapsto y$ is proper, we can then apply again [17, Theorem 8.2.14] and we have that is contained in

The first set is given by

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

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

Therefore

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 [17, Theorem 8.2.9], is a subset of

$$\begin{aligned} \begin{aligned} \Big \{&(x_1, y_1, x_2, y_2, \ldots , x_n, y_n, \xi _1, \eta _1, \xi _2, \eta _2, \ldots , \xi _n, \eta _n) \ | \ \exists I \subseteq \{1, \ldots , n\}, I \ne \varnothing \ \text {s.t.} \\&(x_i, y_i, \xi _i, \eta _i) \in \text {WF}({\mathcal {R}}_A) \ \forall i \in I\ \text {and}\ (x_j, y_j, \xi _j, \eta _j) = (x_j, y_j, 0,0) \\&\text {with} \ (x_j, y_j)\in {\text {supp} }(r_A) \ \forall j \in \{1, \ldots , n\}\setminus I \Big \}\ . \end{aligned} \end{aligned}$$

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 [17, Theorem 8.2.13] we know that the composition is well-defined if $\text {WF}(F^{(n)}_{{\mathcal {R}}_A u}) \cap \text {WF}'({{\mathcal {R}}_A}^{\wedge n})_{M^n}= \varnothing$; but

$$\begin{aligned} \left\{ (x_1, \ldots , x_n, \xi _1, \ldots , \xi _n) \ | \ (x_1, y_1, \ldots , x_n, y_n, {-}\xi _1, 0, \ldots , {-}\xi _n, 0) {\in } \text {WF}({{\mathcal {R}}_A}^{\wedge n})\right\} {=} \varnothing \ . \end{aligned}$$

We also infer that

$$\begin{aligned} \text {WF}\left( ({\mathscr {R}}_A(F))_u^{(n)}\right) \subseteq \text {WF}({{\mathcal {R}}_A}^{\wedge n})_{M^n} \cup \text {WF}'({{\mathcal {R}}_A}^{\wedge n}) \circ \text {WF}(F^{(n)}_{{\mathcal {R}}_A u}) \end{aligned}$$

i.e.

$$\begin{aligned} \begin{aligned} \text {WF}\left( ({\mathscr {R}}_A(F))_u^{(n)}\right) \subseteq \Big \{&(x_1, \ldots , x_n, \xi _1, \ldots , \xi _n) \ | \\&(x_1, y_1, \ldots , x_n, y_n, \xi _1, -\eta _1, \ldots , \xi _n, -\eta _n) \in \text {WF}({\mathcal {R}}_A^{\wedge n})\\&\text {for some} \ (y_1, \ldots , y_n, \eta _1, \ldots , \eta _n)\in \text {WF}(F_{{\mathcal {R}}_A u}^{(n)})\Big \}\ . \end{aligned} \end{aligned}$$

Using this, we shall prove (5.5) by contradiction. Assume then that $(x_1, \ldots , x_n, \xi _1, \ldots , \xi _n)\in \overline{V_{x_1}^+} \times \cdots \times \overline{V_{x_n}^+}$; then there exists $(y_1, \ldots , y_n, \eta _1, \ldots , \eta _n)\in \text {WF}(F_{{\mathcal {R}}_A u}^{(n)})$ such that

$$\begin{aligned} (x_1, y_1, \ldots , x_n, y_n, \xi _1, -\eta _1, \ldots , \xi _n, -\eta _n)\in \text {WF}({{\mathcal {R}}_A}^{\wedge n})\ . \end{aligned}$$

If $i\in I$, then $(x_i, y_i, \xi _i, -\eta _i) \in \text {WF}({{\mathcal {R}}_A})$, that is, either

or

$$\begin{aligned} x_i = y_i \qquad \eta _i =\xi _i\ . \end{aligned}$$

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

$$\begin{aligned} (x_i, y_i, \xi _i, \eta _i) = (x_i, y_i, 0, 0) \end{aligned}$$

i.e. $\eta _i = 0 \in \overline{V^+_{y_i}}$. Thus, we conclude that $(y_1, \ldots , y_n, \eta _1, \ldots , \eta _n)\in \overline{V^+_{y_1}} \times \cdots \overline{V^+_{y_n}}$; but we reached a contradiction, as this is not possible by the definition of F. The case $(x_1, \ldots , x_n, \xi _1, \ldots , \xi _n)\in \overline{V_{x_1}^-} \times \cdots \times \overline{V_{x_n}^-}$ 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

or

$$\begin{aligned} x_i = y_i \qquad \xi _i = \eta _i\ . \end{aligned}$$

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 $(y_1, \ldots , y_n, \eta _1, \ldots , \eta _n)\in \overline{V^-_{y_1}} \times \cdots \overline{V^-_{y_n}}$, which is again a contradiction.

We thus have the well-posedness of the map ${\mathscr {R}}_A :{\mathcal {A}}(S_G)[\![\hbar ]\!]\rightarrow {\mathcal {A}}(S)[\![\hbar ]\!]$. Recall that $R_A$ admits an inverse which is explicitly given by (3.9); it can be shown that by defining an analogous $\widehat{{\mathscr {R}}}_A :{\mathcal {A}}(S)[\![\hbar ]\!] \rightarrow {\mathcal {A}}(S_G)[\![\hbar ]\!]$ we reach the same conclusion, and that ${\mathscr {R}}_A \circ \widehat{{\mathscr {R}}}_A = \text {id}_{{\mathcal {A}}(S)[\![\hbar ]\!]}$ and $\widehat{{\mathscr {R}}}_A \circ {\mathscr {R}}_A = \text {id}_{{\mathcal {A}}(S_G)[\![\hbar ]\!]}$. 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 $\Gamma ^n_\zeta (G,F)$ appearing in the sum can be formally written, on an homogeneous element as

$$\begin{aligned}{} & {} \Gamma ^n_\zeta (G,F)(u_1 \wedge \cdots \wedge u_{p+q-2n}) = \left( \frac{i}{2}\right) ^n\sum _{\sigma \in S_{p+q-2n}}\text {sgn}(\sigma ) \\{} & {} \qquad \times \int _{M^{2n}}d\mu _g(x_1)d \mu _g(y_1) \cdots d\mu _g(x_n) d\mu _g(y_n) \\{} & {} \qquad \left( F^{(n)}_{u_{\sigma (q-n+1)} \wedge \cdots \wedge u_{\sigma (p+q-2n)}}\right) _{f_1 \cdots f_n}(y_1, \ldots , y_n)\left( G^{(n)}_{u_{\sigma (1)} \wedge \cdots \wedge U_{\sigma (q-n)}}\right) _{g_1 \cdots g_n}\\ {}{} & {} \qquad (x_1, \ldots , x_n)\\{} & {} \qquad \zeta _{s_1 t_1}(x_1, y_1) \cdots \zeta _{s_nt_n}(x_n, y_n)h^{f_1s_1}h^{t_1g_1}\cdots h^{f_ns_n}h^{t_ng_n}\ . \end{aligned}$$

We are interested in computing $\Gamma _\zeta ^n({\mathscr {R}}_A(G), {\mathscr {R}}_A(F))$ 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

$$\begin{aligned}{} & {} \int _{M^{2n}}d\mu _g(\xi _1)d\mu _g(\eta _1)\cdots d\mu _g(\xi _n)d\mu _g(\eta _n) d\mu _g(x_1)d \mu _g(y_1) \cdots d\mu _g(x_n) d\mu _g(y_n)\\{} & {} \qquad \left( F^{(n)}_{{{\mathcal {R}}_A}^{\wedge p-n}(u_{\sigma (q-n+1)} \wedge \cdots \wedge u_{\sigma (p+q-2n)})}\right) _{l_1 \cdots l_n}(\xi _1, \ldots , \xi _n)\\{} & {} \qquad \left( G^{(n)}_{{{\mathcal {R}}_A}^{\wedge q-n}(u_{\sigma (1)} \wedge \cdots \wedge u_{\sigma (q-n)})}\right) _{k_1 \cdots k_n}(\eta _1, \ldots , \eta _n)\\{} & {} \qquad {r_A}_{f_1 u_1}(x_1, \xi _1) {r_A}_{g_1 v_1}(y_1, \eta _1) \cdots {r_A}_{f_n u_n}(x_n, \xi _n) {r_A}_{g_n v_n}(y_n, \eta _n) h^{l_1 u_1}h^{v_1 k_1} \cdots \\ {}{} & {} \qquad h^{l_n u_n}h^{v_n k_n}\zeta _{s_1 t_1}(x_1, y_1) \cdots \zeta _{s_nt_n}(x_n, y_n) h^{f_1s_1}h^{t_1g_1}\cdots h^{f_ns_n}h^{t_ng_n}\ . \end{aligned}$$

By performing a simple computation one can notice that

$$\begin{aligned} \begin{aligned}&\int _{M^2}d\mu _g(x_i)d\mu _g(y_i) {{r}_A}_{f_i u_i}(x_i, \xi _i) {{r}_A}_{g_i v_i}(y_i, \eta _i) \zeta _{s_i t_i}(x_i, y_i) h^{f_i s_i} h^{t_i g_i} \\&\qquad = \int _{M^2}d\mu _g(x_i)d\mu _g(y_i){{r}_A}_{ u_i f_i}^*(\xi _i,x_i) {{r}_A}_{v_i g_i}^*(\eta _i, y_i) \zeta _{s_i t_i}(x_i, y_i) h^{f_i s_i} h^{t_i g_i}\\&\qquad = {\zeta _G}_{u_i v_i }(\xi _i, \eta _i) \end{aligned} \end{aligned}$$

and therefore we arrive at

$$\begin{aligned} \begin{aligned}&\int _{M^{2n}}d\mu _g(\xi _1)d\mu _g(\eta _1)\cdots d\mu _g(\xi _n)d\mu _g(\eta _n)\\&\qquad \left( F^{(n)}_{{{\mathcal {R}}_A}^{\wedge p-n}(u_{\sigma (q-n+1)} \wedge \cdots \wedge u_{\sigma (p+q-2n)})}\right) _{l_1 \cdots l_n}(\xi _1, \ldots , \xi _n)\\&\qquad \left( G^{(n)}_{{{\mathcal {R}}_A}^{\wedge q-n}(u_{\sigma (1)} \wedge \cdots \wedge u_{\sigma (q-n)})}\right) _{k_1 \cdots k_n}(\eta _1, \ldots , \eta _n)\\&\qquad {\zeta _G}_{u_1 v_1}(\xi _1, \eta _1) \cdots {\zeta _G}_{u_n v_n}(\xi _n, \eta _n)h^{l_1 u_1}h^{v_1 k_1} \cdots h^{l_n u_n}h^{v_n k_n} \end{aligned} \end{aligned}$$

which entails that

$$\begin{aligned} \Gamma ^n_\zeta ({\mathscr {R}}_A(G), {\mathscr {R}}_A(F))(u_1 \wedge \cdots \wedge u_{p+q-2n}) = {\mathscr {R}}_A(\Gamma ^n_{\zeta _G}(G,F))(u_1 \wedge \cdots \wedge u_{p+q-2n})\ . \end{aligned}$$

The result can be extended to the whole 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,

$$\begin{aligned} {\mathscr {R}}_A(F) \star {\mathscr {R}}_A(H) = {\mathscr {R}}_A(F \star _G H) \end{aligned}$$

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 or ) 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

$$\begin{aligned} \begin{aligned} ({\mathscr {R}}_A(F))^*(u)&= \overline{({\mathscr {R}}_A(F)) (u^*)} = \sum _{p\in {\mathbb {N}}} \overline{{\langle {F_p, {{\mathcal {R}}_A}^{\wedge p}u_p^*}\rangle }}\ . \end{aligned} \end{aligned}$$

Now, if $u_p = u_{i_1} \wedge \cdots \wedge u_{i_p}$, we have that

$$\begin{aligned} {{\mathcal {R}}_A}^{\wedge p}(u_p^*){} & {} = {\mathcal {R}}_A u_{i_p}^*\wedge \cdots \wedge {\mathcal {R}}_A u_{i_1}^*= ({\mathcal {R}}_A u_{i_p})^*\wedge \cdots \wedge ({\mathcal {R}}_A u_{i_1})^*\\ {}{} & {} = \left( {{\mathcal {R}}_A}^{\wedge p} (u_1 \wedge \cdots \wedge u_p)\right) ^*\ . \end{aligned}$$

By continuity then it holds that

$$\begin{aligned} \sum _{p\in {\mathbb {N}}} \overline{{\langle {F_p, {{\mathcal {R}}_A}^{\wedge ^p}u_p^*}\rangle }} = \sum _{p\in {\mathbb {N}}} \overline{{\langle {F_p, ({{\mathcal {R}}_A}^{\wedge p}u_p)^*}\rangle }} = ({\mathscr {R}}_A(F^*))(u)\ . \end{aligned}$$

Therefore, ${\mathscr {R}}_A:{\mathcal {A}}(S_G(M))[[\hbar ]]\rightarrow {\mathcal {A}}(S(M))[[\hbar ]]$ is a well-defined algebraic $*$-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. $\square$

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 contractible spacetimes, which rules out topological effects (i.e. inequivalent spinor structures [40] and Aharonov–Bohm like effects, for which see, e.g., [41, 42]), the second is our assumption about the past-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 [43] 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 [44].

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., [33]) 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 [14] from first principles, avoiding ad hoc assumptions and rigorously controlling eventual approximations.