In this appendix, we will give some details on the derivation of (4.60).

We start by considering the retarded and advanced Green operators for the operator T given in (4.53). Decomposing \(\Gamma ^\infty (\Lambda ^1M\oplus B)\cong \Gamma ^\infty (\Lambda ^1M) \oplus \Gamma ^\infty (\Lambda ^1M)\oplus \Gamma ^\infty (\Lambda ^0M)\), we can write

$$\begin E^\pm _&=\begin \tilde_^\pm & \tilde_^\pm & \tilde_^\pm \\ \tilde_^\pm & \tilde_^\pm & \tilde_^\pm \\ \tilde_^\pm & \tilde_^\pm & \tilde_^\pm \end\, . \end$$

(A.1)

Plugging this decomposition into (4.54) and in turn into (4.59), we find that the advanced and retarded Green operators for the coupled Proca–scalar operator P (4.48) can be written as

$$\begin E^\pm _P=\begin \tilde^\pm _(1-m^\textrm\delta ) & \tilde_^\pm +\tilde^\pm _ \textrm\\ \tilde^\pm _(1-m^\textrm\delta ) & \tilde_^\pm +\tilde^\pm _ \textrm\end\, . \end$$

(A.2)

To analyse this further, we turn to \(E^\pm _T\). Using the same decomposition as before, we can use the properties G1 and G2 of \(E^\pm _T\) together with the explicit form of T to derive the following equations:

From G1, we obtain the following identities on compactly supported sections of \(\Lambda ^1M\) or \(\Lambda ^0M\)

$$\begin v^\bullet \tilde^\pm _&=K^_}\tilde^\pm _\, ,\end$$

(A.3)

$$\begin v^\bullet \tilde^\pm _&=K^_}\tilde^\pm _\, ,\end$$

(A.4)

$$\begin \textrmv^\bullet \tilde^\pm _&=K^_}\tilde^\pm _\, ,\end$$

(A.5)

$$\begin \textrmv^\bullet \tilde^\pm _&=K^_}\tilde^\pm _ \end$$

(A.6)

and acting from the left on the first two with \(E^\pm __}}\) and the second two with \(E^\pm __}}\), we can express \(\tilde^\pm _\), \(\tilde^\pm _\), \(\tilde^\pm _\), and \(\tilde^\pm _\) in terms of \(\tilde^\pm _\), \(\tilde^\pm _\), \(\tilde^\pm _\), v, differential operators, and Green operators for the Klein–Gordon operator (we also use the compact support of v and the support property G3). From G2, we obtain the additional identities

$$\begin -m^\tilde^\pm _\textrmv^\bullet&=\tilde^\pm _K^_}\, , \end$$

(A.7)

$$\begin -\tilde^\pm _(v-m^ \textrm(\delta v))&=\tilde^\pm _K^_} \end$$

(A.8)

on sections with compact support. They can be composed with \(E^\pm __}}\) or \(E^\pm __}}\) from the right respectively, to obtain expressions for \(\tilde^\pm _\) and \(\tilde^\pm _\). Furthermore, we take from G1 the identities

$$\begin K^_\tilde^\pm _-\textrmv^\bullet \tilde^\pm _&=\textrm\, ,\end$$

(A.9)

$$\begin K^_\tilde^\pm _-v^\bullet \tilde^\pm _&=\textrm\, ,\end$$

(A.10)

$$\begin m^\textrmv^\bullet \tilde^\pm _+K^_m\tilde^\pm _+(v-m^\textrm(\delta v))\tilde^\pm _&=\textrm\, , \end$$

(A.11)

and from G2 the identity

$$\begin \tilde^\pm _K^_-\tilde^\pm _\textrmv^\bullet -\tilde^\pm _v^\bullet =\textrm\, . \end$$

(A.12)

Plugging the results for the off-diagonal components of \(E^\pm _T\) into the first two and acting with \(E^\pm __}\) or \(E^\pm __}\) from the left, respectively, we obtain expressions for \(\tilde^\pm _\) and \(\tilde^\pm _\). The third equation from G1 and the equation for G2, combined with the previous result, gives (4.62).

In the last step, we use the fact that \(E^\pm _} \textrm=\textrmE^\pm _}\) and \(\textrmv^\bullet \textrm-\textrm(\delta v)=-\textrm\delta v\) as an operator acting on 0-form fields. The same identities can be used to compute

$$\begin \tilde^\pm _+\tilde^\pm _\textrm&=-\tilde^\pm _(\textrm-m^\textrm\delta ) v E^\pm __}\end$$

(A.13)

$$\begin \tilde^\pm _+\tilde^\pm _ d&= E^\pm __}}-E^\pm __}v^\bullet \tilde^\pm _(\textrm-m^\textrm\delta )v E^\pm __}\end$$

(A.14)

$$\begin \tilde^\pm _(1-m^\textrm\delta )&=E^\pm __}v^\bullet \tilde^\pm _(\textrm-m^\textrm\delta )\, . \end$$

(A.15)

Plugging these results into (A.2), one obtains (4.60).

In this section, we will make precise in which sense the higher-order terms in (6.48) are of order \(\lambda ^4\). As mentioned there, it is enough to show that \(\mathcal \overline^} \rho u_\theta ^\bullet E^-_\rho u_\theta E^-__}f}\) converges in \(_W\) as \(\lambda \rightarrow 0\). As in Sect. 6, we will drop the superscripts on \(K_\) and \(K_m\), since only \(K_^\) and \(K_m^\) play a role in this derivation.

We note that the only \(\lambda \)-dependence is contained in the operator \(E^-_\). We will make use of various facts that will be proved below. First, we claim that

$$\begin \Vert \mathcal u_\theta h\Vert __W}\le C \Vert h\Vert _}(M)} \end$$

(B.1)

for all compact \(\tilde\) and \(h\in C^\infty _}(M)\). Here and below, \(H^s_}(M)\) is the Sobolev space of order s of functions supported in \(\tilde\), while \(H^s_}(M,\Lambda ^1M)\) is the analogous Sobolev space of 1-forms. (Our definitions follow those in the Appendix of [16].)

Second, if \(\tilde\) contains \(\,}}\rho \) then \(\rho E^-_}\) defines a bounded operator between \(H^2_}(M)\) and \(H^3_}(M)\). Third, we will show that \(\rho u_\theta ^\bullet E^-_\) defines a bounded operator \(\mathcal (H^3_}(M;\Lambda ^1M),H^_}(M))\), converging to \(\rho u_\theta ^\bullet E^-_}}\) in operator norm as \(\lambda \rightarrow 0\). Here \(E^-_}} =E^-_ (1-m^\textrm\delta )\) is the advanced Green operator for the real, uncoupled Proca field. Fourth, for each \(f\in C_0^\infty (M)\) one has \(\rho u_\theta E^-_}f\in \Gamma _}^\infty (\Lambda ^1M) \subset H^3_}(M;\Lambda ^1M)\). Given these facts, it follows that

$$\begin \mathcal \overline} \rho u_\theta ^\bullet E^-_\rho u_\theta E^-_}f} \rightarrow \mathcal \overline} \rho u_\theta ^\bullet E^-_}}}\rho u_\theta E^-_}f} \end$$

(B.2)

in \(_W\) as \(\lambda \rightarrow 0\) for each fixed \(f\in C_0^\infty (M)\), which is our required overall result.

It remains to establish the four facts above. Indeed, the second is a standard fact about Green operators of normally hyperbolic operators, which gain one derivative by the results of [14, Theorem 6.5.3] (see also [36] for a bundle version). Meanwhile the fourth fact is trivial so only the first and third need be shown.

Starting with the first fact, let \(h\in C^\infty _}(M)\) and consider \(\mathcal u_\theta h\). Then

$$\begin u_\theta h \right\| }__W}^2=\int \limits _^3}\frac^3}}})} \phi _\theta (})(\omega (}),}) \right| }^2\, , \end$$

(B.3)

where

$$\begin \phi _\theta (})= & -\eta ^(\Pi u_\theta , \Pi u_\theta )=1+m^} \right| }^2\sin ^2\alpha \cos ^2(\beta -\theta ) \nonumber \\ \le & 1+m^} \right| }^2 = m^\omega (})^2 \end$$

(B.4)

can be computed using the parametrization (6.25) of the 4-covector \(k_\mu \). We can estimate (B.3) by

$$\begin u_\theta h \right\| }__W}^2&\le \frac\int \limits _^3}\frac^3}}})^5} })^3\widehat(\omega (}),}) \right| }^2\,\nonumber \\&= \frac\int \limits _^3}\frac^3}}})^5} (\omega (}),}) \right| }^2\,\nonumber \\&\le C _^4)}^2\, , \end$$

(B.5)

for some constant \(C>0\), where we have used the notation t for the \(x^0\)-coordinate. In the last step, we have used the facts that \(\omega ^\in L^1(\mathbb ^3,\textrm^})\) and that the \(L^\infty \)-norm of a function’s Fourier transform is bounded by its \(L^1\)-norm. Note that one could make do with a smaller number of derivatives, but this result is sufficient for our purposes. Finally, let \(\chi \in L^2(\mathbb ^4)\) so that \(\chi (x)=1\) on \(\tilde\). Then we have

$$\begin _^4)}&=_^4)}\le _^4)}_^4)} \le C' _}(M)} , \end$$

(B.6)

and combining with the previous estimate the first fact is established.

Lastly, recall that \(E^\pm _=\tilde^_(1-m^\textrm\delta )\) by (4.61), so the \(\lambda \)-dependence of \(E^\pm _\) is contained in \(\tilde^_\). The operators \(\tilde^\pm _\) satisfy the equations (4.62), i.e., they invert the non-local operator \(O^\pm =K_m+\lambda ^2(\textrm-m^\textrm\delta )v_\theta E^\pm _}v_\theta ^\bullet \) where \(v_\theta =\rho u_\theta \in \Gamma _0^\infty (\Lambda ^1M)\) is a fixed, real one-form. Acting on the equation \(\tilde^\pm _O^\pm =\textrm\) with \(E^\pm _\) from the right leads to the equation

$$\begin \tilde^\pm _(\textrm+\lambda ^2 (\textrm-m^\textrm\delta )v_\theta E^\pm _} v_\theta ^\bullet E^\pm _)=E^\pm _\, , \end$$

(B.7)

which holds on \(\Gamma _0^\infty (\Lambda ^1M)\).

Now the operator \((\textrm-m^\textrm\delta )v_\theta E^\pm _} v_\theta ^\bullet E^\pm _\) extends to a bounded operator on the Hilbert space \(H^s_} (M;\Lambda ^1M)\) for any \(s\in \mathbb \) and any compact set \(\tilde\subset M\) that contains \(\,}}\rho \). Consequently, the operator

$$\begin (\textrm+\lambda ^2 (\textrm-m^\textrm\delta )v_\theta E^\pm _} v_\theta ^\bullet E^\pm _)^: H^s_}(M;\Lambda ^1M)\rightarrow H^s_}(M;\Lambda ^1M) \end$$

(B.8)

is the resolvent of a bounded operator, which is well-defined and bounded if \(|\lambda |<\lambda _0\) for some \(\lambda _0>0\) which may depend on s; furthermore the resolvent is holomorphic in \(\lambda \) within this disk.

From now on, let us fix some arbitrary \(s\in \mathbb \) and let us assume that \(0<\lambda <\lambda _0\), so that the resolvent is a bounded operator on \(H^s_}(M;\Lambda ^1M)\). Then acting on (B.7) with the resolvent from the right, and by contraction with the compactly supported 1-form \(v_\theta \) from the left, one obtains the formula

$$\begin v_\theta ^\bullet \tilde^\pm _= v_\theta ^\bullet E^\pm _ (\textrm+\lambda ^2 (\textrm-m^\textrm\delta )v_\theta E^\pm _} v_\theta ^\bullet E^\pm _)^\, \end$$

(B.9)

on \(\Gamma ^\infty _}(\Lambda ^1M)\subset H^s_}(M;\Lambda ^1M)\). By the preceding discussion, we can thus see that \(v_\theta ^\bullet \tilde^\pm _\) extends to a bounded operator from \(H^s_}(M;\Lambda ^1M)\) to \(H^_}(M)\), and that the \(\lambda \)-dependence of this operator is determined by the \(\lambda \)-dependence of the resolvent. Consequently, as \(\lambda \rightarrow 0\), \(v_\theta ^\bullet \tilde^\pm _\) converges in norm to \(v_\theta ^\bullet E^\pm _ \in \mathcal (H^s_}(M;\Lambda ^1M),H^_}(M))\). Recalling (4.61), we notice that no additional \(\lambda \)-dependence is introduced in going from \(\tilde^\pm _\) to \(E^\pm _ =\tilde^\pm _(1-m^\textrm\delta )\), but that the transition loses up to two derivatives. We conclude that, for any \(s\in \mathbb \), \(v_\theta ^\bullet E^-_:\, H^s_}(M;\Lambda ^1M)\rightarrow H^_}(M)\) is a bounded operator, which converges in norm to \(v_\theta ^\bullet E^-_}}\) in \(\mathcal (H^s_}(M;\Lambda ^1M),H^_}(M))\). While one can make stronger statements concerning differentiability in \(\lambda \), the convergence result is all that is needed for our purposes. The third fact above is the special case \(s=3\).

In this section, we will show that the factor \(A(s,\rho , f)\) also arises in the response of our probe field when it is linearly coupled to a scalar field.

For this, we consider as our system theory a complex scalar of mass m (i.e., the same mass as the Proca field in Sect. 6). The uncoupled theory is then described by \(K_m\oplus K_\) (we drop the (0)-superscript on the Klein-Gordon operator since we will only deal with scalar functions in this section), and we assume that the differential operator describing the coupled theory takes the form

$$\begin P=\begin K_m & \lambda \rho \\ \lambda \rho & K_\end\, , \end$$

(C.1)

where \(\lambda \in \mathbb \) and \(\rho \in C^\infty _0(M;[0,1])\) are the same coupling constant and scalar test function as considered for the Proca-scalar system. The probe observable we consider for this system is

$$\begin O_f=\Phi (f)^*\Phi (f)-c_f1\!\!1_\mathscr \, , \end$$

(C.2)

where \(f\in C_0^\infty (M^+)\) is the same test function as in the Proca-scalar system, and \(c_f\in \mathbb \) is a ‘calibration constant’ which may depend on f, \(\lambda \), and \(\rho \).

A calculation along the same lines as in Sect. 6 then shows that the induced observable in this case is given by

$$\begin \mathcal _(O_f)=\Psi (h^-)^*\Psi (h^-)+\left( \Omega _\Phi (\Phi (f^-)^*\Phi (f^-))-c_f\right) 1\!\!1_\mathscr \, , \end$$

(C.3)

where \(\mathscr \) is the algebra of observables of the system, and \(\Psi (f)\), \(\Psi (f)^*\) are its generators. We have also defined

$$\begin \begin h^-\\ f^- \end=\begin -\lambda \rho E^-_} f+\mathcal (\lambda ^3) \\ f+\lambda ^2\rho E^-_\rho E^-_}f+\mathcal (\lambda ^4) \end\, , \end$$

(C.4)

which follows from a Born expansion for the coupled propagator, see for example [25] or [21].

As before, we consider system and probe in states in the folium of the Minkowski vacuum, so that the probe fields \(\Phi (f)\) are represented as before, and the system fields \(\Psi \) are represented in the same way as the probe fields with \(\) replaced by m.

We then select the system preparation state to be the state

$$\begin \omega _(A)=\frac__\Psi }\, , \end$$

(C.5)

where \(\mathcal _\Psi \), \(\Omega _\Psi \in \mathcal _\Psi \), \(\pi _\Psi \), and \(a^*_\Psi \) are the Hilbert space, Minkowski vacuum state, representation of \(\mathscr \) on \(\mathcal _\Psi \), and creation operator for the complex scalar field \(\Psi \), respectively. The function \(s\in L^2(\mathbb ^3, (2\pi )^\textrm^3})\) is chosen to be the same as in the scalar-Proca system. Thus this is an n-particle single-mode state with momentum concentrated around the z-axis.

Using the commutation relations of the creation and annihilation operators, we then obtain

$$\begin \omega _(\mathcal _(O_f))=n_\Psi \overline, s \right\rangle }__\Psi } \right| }^2+_\Psi h^- \right\| }__\Psi }^2+_\Phi f^- \right\| }__\Phi }^2-c_f\, . \end$$

(C.6)

By setting \(c_f=_\Psi h^- \right\| }__\Psi }^2+_\Phi f^- \right\| }__\Phi }^2\), which is independent of the choice of s and n, and using the Born expansion for \(h^-\), we obtain

$$\begin \omega _(\mathcal _(O_f))&=\lambda ^2n^3}} s(})\overline_\Psi (\rho E^-_} f)}(}) \right| }^2+\mathcal (\lambda ^3)\end$$

(C.7)

$$\begin&=\lambda ^2n^3}}}))^} s(})\widehat} f}(-\omega (}),-}) \right| }^2+\mathcal (\lambda ^3)\,. \end$$

(C.8)

Comparing with (6.53), one can see immediately that this is indeed equal to \(\lambda ^2^2+\mathcal (\lambda ^3)\).