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Asymptotic Morphisms and Superselection Theory in the Scaling Limit II: Analysis of Some Models

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Abstract

We introduced in a previous paper a general notion of asymptotic morphism of a given local net of observables, which allows to describe the sectors of a corresponding scaling limit net. Here, as an application, we illustrate the general framework by analyzing the Schwinger model, which features confined charges. In particular, we explicitly construct asymptotic morphisms for these sectors in restriction to the subnet generated by the derivatives of the field and momentum at time zero. As a consequence, the confined charges of the Schwinger model are in principle accessible to observation. We also study the obstructions, that can be traced back to the infrared singular nature of the massless free field in \(d=2\), to perform the same construction for the complete Schwinger model net. Finally, we exhibit asymptotic morphisms for the net generated by the massive free charged scalar field in four dimensions, where no infrared problems appear in the scaling limit.

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Notes

  1. Where this means of course that one has to disregard the fact that performing measurements at arbitrarily small length scales requires unbounded energies.

  2. A representation of \({\mathfrak {W}}\) in which \(\tau ^{(0)}\) is unitarily implemented on a separable Hilbert space is constructed in [25].

  3. The results in [12] also suggest that the scaling limit of the C*-subalgebra of the scaling algebra generated by (smoothed-out) functions \(\lambda \mapsto W(\delta _\lambda f)\) and \(\lambda \mapsto W(|\log \lambda |^{1/2}\delta _\lambda f)\) is isomorphic to \(\mathcal{A}^{(0)}\otimes \mathcal{Z}\), with \(\mathcal{Z}\) (a subalgebra of) the center of \({\mathcal {A}}_{0,\iota }^{(m)}\).

  4. Note that if \({{\text {supp}}\,}f\subset O\), since O is open it is always possible to choose \({{\text {supp}}\,}h\) so small that \(\underline{\alpha }_h{\underline{W}}(f) \in \underline{\mathfrak {A}}^{(m)}(O)\).

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Acknowledgements

We are grateful to D. Buchholz and to D. Guido for useful discussions about the subject of this work and to D. Buchholz also for his comments on a preliminary version, and we thank E. Valdinoci for pointing out reference [35]. R. C. is partially supported by the Sapienza Ricerca Scientifica 2017 grant “Algebre di Operatori e Analisi Armonica Noncommutativa”. G. M. is partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, the ERC Advanced Grant 669240 “Quantum Algebraic Structures and Models”, the INDAM-GNAMPA, and the Tor Vergata University grant “Operator Algebras and Applications to Noncommutative Structures in Mathematics and Physics”.

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On the Quasiequivalence of Vacuum States of Different Masses in 2D

On the Quasiequivalence of Vacuum States of Different Masses in 2D

Recall that in Sect. 3 we defined the von Neumann algebras

$$\begin{aligned} \mathcal{C}^{(m)}(O_I) := \Big \{ \pi ^{(m)}(W(f))\,:\, {\mathrm{supp}}\, f \subset I, \, {\textstyle \int _I f} = 0\Big \}'', \quad m\ge 0 \ . \end{aligned}$$

In this appendix we provide a proof of the following result.

Theorem A.1

For each bounded interval \(I \subset {\mathbb {R}}\) there exists a von Neumann algebra isomorphism \(\phi \) between \(\mathcal{C}^{(m)}(O_I)\), \(m>0\), and \(\mathcal{C}^{(0)}(O_I)\) such that \(\phi (\pi ^{(m)}(W(f))) = \pi ^{(0)}(W(f))\).

To show this fact, we make appeal to the general results of [4]. To make contact with the formalism employed there in the description of CCR algebras, we introduce then the notation

$$\begin{aligned} \mathcal{D}_0(I) := \Big \{ f \in \mathcal{D}(I)\,:\, {\textstyle \int _I f } = 0\Big \}, \end{aligned}$$

considered as a real vector space, and we define the complex vector space \(K := \mathcal{D}_0(I) \oplus \mathcal{D}_0(I)\) with complex structure defined by

$$\begin{aligned} i \cdot (f_1\oplus f_2) := (-f_2)\oplus f_1. \end{aligned}$$

Moreover, we define a conjugate linear involution \(\Gamma : K \rightarrow K\) and an indefinite inner product \(\gamma : K \times K \rightarrow {\mathbb {C}}\) as

$$\begin{aligned} \Gamma (f_1 \oplus f_2)&:= f_1\oplus (-f_2),\\ \gamma (f_1 \oplus f_2,g_1\oplus g_2)&:= \frac{1}{2}\big (\langle f_1+if_2,g_1+ig_2\rangle - \langle g_1-ig_2,f_1-if_2\rangle \big ), \end{aligned}$$

with \(\langle \cdot , \cdot \rangle \) the standard scalar product on \(L^2({\mathbb {R}})\). They satisfy \(\gamma (\Gamma x, \Gamma y) = -\gamma (y,x)\) for all \(x, y \in K\). To these data, [3, Sec. 2] associates the self-dual CCR algebra \({\mathfrak {A}}(K,\gamma ,\Gamma )\). This is the \(*\)-algebra generated by symbols \(B(f_1 \oplus f_2)\), \(f_1 \oplus f_2 \in K\), satisfying the natural relations suggested by the identification of \(B(f_1 \oplus f_2)\) with the Fock space operators \(\varphi (f_1) + i \varphi (f_2)\), where \(\varphi (f) = \phi ({{\mathrm {Re}}\,}f) - \pi ({{\mathrm {Im}}\,}f) = \frac{1}{\sqrt{2}} \big [ a(\omega _m^{-1/2} {{\mathrm {Re}}\,}f + i \omega _m^{1/2} {{\mathrm {Im}}\,}f) + a(\omega _m^{-1/2} {{\mathrm {Re}}\,}f + i \omega _m^{1/2} {{\mathrm {Im}}\,}f)^* \big ]\). Notice that \(\pi ^{(m)}(W(f)) = e^{i \varphi (f)}\) for \(m>0\), while \(\pi ^{(0)}(W(f)) = e^{i \varphi (f)} \otimes \mathbf{1}\) acts on the tensor product of the (Bosonic) Fock space on \(\mathcal{D}_0({\mathbb {R}})\) with a nonseparable multiplicity space.

We now introduce the \({\mathbb {R}}\)-bilinear map \(\langle \cdot , \cdot \rangle _m : \mathcal{D}_0({\mathbb {R}}) \times \mathcal{D}_0({\mathbb {R}}) \rightarrow {\mathbb {C}}\), \(m \ge 0\), defined by

$$\begin{aligned}\begin{aligned} \langle f,g\rangle _m&:= \frac{1}{2} \int _{\mathbb {R}}d{\varvec{p}}\overline{\left[ \omega _m({\varvec{p}})^{-1/2} ({{\mathrm {Re}}\,}f)^{\wedge }({\varvec{p}}) + i\omega _m({\varvec{p}})^{1/2} ({{\mathrm {Im}}\,}f)^{\wedge }({\varvec{p}})\right] } \\&\quad \times \big [\omega _m({\varvec{p}})^{-1/2} ({{\mathrm {Re}}\,}g)^{\wedge }({\varvec{p}}) + i\omega _m({\varvec{p}})^{1/2} ({{\mathrm {Im}}\,}g)^{\wedge }({\varvec{p}})\big ]\,. \end{aligned}\end{aligned}$$

It is clear that \({{\mathrm {Re}}\,}\langle \cdot , \cdot \rangle _m\) is a real scalar product inducing the norm \(\Vert \cdot \Vert _m\) on \(\mathcal{D}_0({\mathbb {R}})\) and it is easy to verify that \({{\mathrm {Im}}\,}\langle f, g \rangle _m = \frac{1}{2} {{\mathrm {Im}}\,}\langle f, g \rangle \). We can then introduce on \( K \times K\) the hermitian form

$$\begin{aligned} S_m(f_1\oplus f_2,g_1\oplus g_2) := \langle f_1,g_1\rangle _m + i \langle f_1,g_2\rangle _m -i \langle f_2,g_1\rangle _m + \langle f_2,g_2\rangle _m. \end{aligned}$$

This is nothing but \(\big \langle \Omega ^{(m)},(\varphi (f_1) + i \varphi (f_2))^* (\varphi (g_1) + i \varphi (g_2))\Omega ^{(m)}\big \rangle \) if \(m>0\) and \(\big \langle \Omega ^{(0)},\big [(\varphi (f_1) + i \varphi (f_2))^* (\varphi (g_1) + i \varphi (g_2))\big ] \otimes \mathbf{1}\, \Omega ^{(0)}\big \rangle \) otherwise.

One verifies that \(S_m(x,x) \ge 0\) and \(S_m(x,y) - S_m(\Gamma y,\Gamma x) = \gamma (x,y)\) for all \(x,y \in K\), and then, by [3, Lem. 3.5], there exists a unique quasi-free state \(\varphi _m= \varphi _{S_m}\) on \({\mathfrak {A}}(K,\gamma ,\Gamma )\) such that \(\varphi _m(x^*y) = S_m(x,y)\).

We now observe that \({{\mathrm {Re}}\,}K := \{x \in K\,:\, \Gamma x = x\} = \mathcal{D}_0(I)\oplus \{0\}\), and that for \(f,g \in \mathcal{D}_0(I)\) we get

$$\begin{aligned} \gamma (f\oplus 0, g\oplus 0) = i \, {{\mathrm {Im}}\,}\langle f,g\rangle , \qquad S_m(f\oplus 0,f\oplus 0) = \Vert f\Vert _m^2. \end{aligned}$$
(A.1)

Then, by [4, Prop. 3.4] and by the separating property of the vacuum vector for \(\mathcal{C}^{(m)}(O_I)\), the Weyl operators \(W_{S_m}(x)\), \(x \in {{\mathrm {Re}}\,}K\), in the GNS representation of \({\mathfrak {A}}(K,\gamma ,\Gamma )\) induced by \(\varphi _m\) generate a von Neumann algebra isomorphic to \(\mathcal{C}^{(m)}(O_I)\) for all \(m \ge 0\).

The next step is to define the scalar product

$$\begin{aligned} \langle x,y\rangle _{S_m} := S_m(x,y) + S_m(\Gamma y, \Gamma x), \qquad x,y \in K, \end{aligned}$$

and to observe that

$$\begin{aligned} \langle f_1\oplus f_2,f_1\oplus f_2\rangle _{S_m} = 2 (\Vert f_1\Vert _m^2 + \Vert f_2\Vert _m^2) = 0 \end{aligned}$$
(A.2)

implies \(f_1 \oplus f_2 = 0\). Then the main Theorem of [4] implies that the statement of Thmorem A.1 holds if and only if:

  1. 1.

    the (Hausdorff) topologies induced on K by \(\langle \cdot , \cdot \rangle _{S_m}\) and \(\langle \cdot , \cdot \rangle _{S_0}\) coincide;

  2. 2.

    denoting by \({\bar{K}}\) the completion of K in the above topology, and by \({\tilde{S}}_m, \tilde{S}_0 : {\bar{K}} \rightarrow {\bar{K}}\) the bounded (by [4, Lem. 3.2]) operators defined by

    $$\begin{aligned} S_m(x,y) = \langle x, \tilde{S}_m y\rangle _{S_m}, \quad S_0(x,y) = \langle x, \tilde{S}_0 y\rangle _{S_m}, \qquad x,y \in K, \end{aligned}$$
    (A.3)

    (with \(\langle \cdot ,\cdot \rangle _{S_m}\) extended by continuity to \({\bar{K}}\)), the operator \(\tilde{S}_m^{1/2}- \tilde{S}_0^{1/2}\) is Hilbert-Schmidt.

The remaining part of this appendix is devoted to showing the validity of 1. and 2.

Proposition A.2

The topologies induced on K by \(\langle \cdot , \cdot \rangle _{S_m}\) and \(\langle \cdot , \cdot \rangle _{S_0}\) coincide.

Proof

By (A.2), it is sufficient to show that the norms \(\Vert \cdot \Vert _m\), \(\Vert \cdot \Vert _0\) are equivalent on \(\mathcal{D}_0(I)\). Now, by the Parseval identity and by the fact that \(\omega _m^{\pm 1/2}\) maps real functions to real functions,

$$\begin{aligned} \begin{aligned} \Vert f\Vert _m^2&= \frac{1}{2} \int _{{\mathbb {R}}} d{\varvec{x}}\left| \left( \omega _m^{-1/2} {{\mathrm {Re}}\,}f + i \omega _m^{1/2}{{\mathrm {Im}}\,}f\right) ({\varvec{x}})\right| ^2 \\&=\frac{1}{2} \int _{\mathbb {R}}d{\varvec{p}}\left[ \omega _m({\varvec{p}})^{-1} |({{\mathrm {Re}}\,}f)^{\wedge }({\varvec{p}})|^2+\omega _m({\varvec{p}}) |({{\mathrm {Im}}\,}f)^{\wedge }({\varvec{p}})|^2\right] , \end{aligned}\end{aligned}$$

and therefore it is sufficient to show that the \(\pm 1\) norms

$$\begin{aligned} \Vert f\Vert _{m,\pm 1}^2 := \int _{\mathbb {R}}d{\varvec{p}}\,\omega _m({\varvec{p}})^{\pm 1} |\hat{f}({\varvec{p}})|^2, \qquad f \in \mathcal{D}_0(I,{\mathbb {R}}), \end{aligned}$$

are respectively equivalent for \(m \ne 0\) and \(m = 0\).

The inequalities \(\Vert f\Vert _{m,-1} \le \Vert f\Vert _{0,-1}\), \(\Vert f\Vert _{0,1} \le \Vert f\Vert _{m,1}\) are obvious. We now prove that \(\Vert f\Vert _{m,1} \le C \Vert f\Vert _{0,1}\) for some \(C > 0\). As in [29], by the inequality

$$\begin{aligned} \omega _m({\varvec{p}}) \le \sqrt{1+m^2}|{\varvec{p}}| + m \chi _{\{|{\varvec{p}}|\le 1\}}({\varvec{p}}) \end{aligned}$$
(A.4)

we get

$$\begin{aligned} \Vert f\Vert ^2_{m,1} \le \sqrt{1+m^2} \Vert f\Vert ^2_{0,1} +m \int _{\mathbb {R}}d{\varvec{p}}|\hat{f}({\varvec{p}})|^2, \end{aligned}$$

and the required inequality is obtained by applying [35, Thm. 1] with \(n=1\), \(p=2\), \(\Omega = I\). Indeed, since \(t \le e^{\pi t}\) for all \(t \ge 0\),

$$\begin{aligned}\begin{aligned} \int _{\mathbb {R}}d{\varvec{p}}|\hat{f}({\varvec{p}})|^2&= \int _I d{\varvec{x}}|f({\varvec{x}})|^2 \le \Vert (-\Delta )^{1/4} f\Vert ^2_{L^2(I)} \int _I d{\varvec{x}}\, e^{\pi \Big |\frac{f({\varvec{x}})}{\Vert (-\Delta )^{1/4} f\Vert _{L^2(I)}}\Big |^2} \\&\le c_{1,2} |I| \,\Vert (-\Delta )^{1/4} f\Vert ^2_{L^2(I)}, \end{aligned}\end{aligned}$$

and moreover there exist \(c' > 0\) such that \(\Vert (-\Delta )^{1/4} f\Vert _{L^2(I)} \le c' \Vert (-\Delta )^{1/4} f\Vert _{L^2({\mathbb {R}})} = c' \Vert f\Vert _{0,1}\), see [35, Rem. 2]. Summing up, we obtain \(\Vert f\Vert _{m,1}^2 \le (\sqrt{1+m^2}+m c_{1,2}\, c'^2\, |I|)\Vert f\Vert _{0,1}^2\).

Finally, we prove \(\Vert f\Vert _{0,-1} \le C \Vert f\Vert _{m,-1}\) for some \(C > 0\). Using (A.4) again, we have

$$\begin{aligned}\begin{aligned} \Vert f\Vert _{0,-1}^2&\le \sqrt{1+m^2} \int _{|{\varvec{p}}| > 1} \frac{d{\varvec{p}}}{\omega _m({\varvec{p}})} |\hat{f}({\varvec{p}})|^2 + \int _{|{\varvec{p}}| \le 1} \frac{d{\varvec{p}}}{|{\varvec{p}}|} |\hat{f}({\varvec{p}})|^2 \\&\le \sqrt{1+m^2} \Vert f\Vert _{m,-1}^2 + \int _{|{\varvec{p}}| \le 1} \frac{d{\varvec{p}}}{|{\varvec{p}}|} |\hat{f}({\varvec{p}})|^2. \end{aligned}\end{aligned}$$

If we now consider a function \(\varphi \in \mathcal{D}({\mathbb {R}})\) such that \(\varphi ({\varvec{x}}) = 1\) for all \({\varvec{x}}\in I\), we have \(f = f \varphi \), and therefore

$$\begin{aligned}\begin{aligned} \int _{|{\varvec{p}}| \le 1} \frac{d{\varvec{p}}}{|{\varvec{p}}|} |\hat{f}({\varvec{p}})|^2&= \int _{|{\varvec{p}}| \le 1} \frac{d{\varvec{p}}}{|{\varvec{p}}|} \left| \int _{\mathbb {R}}d{\varvec{q}}\hat{f}({\varvec{q}})\hat{\varphi }({\varvec{p}}- {\varvec{q}}) \right| ^2 \\&= \int _{|{\varvec{p}}| \le 1} d{\varvec{p}}|{\varvec{p}}| \left| \int _{\mathbb {R}}d{\varvec{q}}\hat{f}({\varvec{q}})\psi ({\varvec{p}},{\varvec{q}}) \right| ^2 \\&= \int _{|{\varvec{p}}| \le 1} d{\varvec{p}}|{\varvec{p}}| \left| \int _{\mathbb {R}}d{\varvec{q}}\frac{{\hat{f}}({\varvec{q}})}{\omega _m({\varvec{q}})^{1/2}}\omega _m({\varvec{q}})^{1/2}\psi ({\varvec{p}},{\varvec{q}}) \right| ^2 \end{aligned}\end{aligned}$$

where in the second equality we have introduced the function \(\psi ({\varvec{p}},{\varvec{q}}) := \frac{{\hat{\varphi }}({\varvec{p}}-{\varvec{q}})-{\hat{\varphi }}(-{\varvec{q}})}{{\varvec{p}}}\) and used the fact that \(\int _{\mathbb {R}}d{\varvec{q}}\hat{f}({\varvec{q}}) \hat{\varphi }(-{\varvec{q}}) = \hat{f}(0) = 0\). From this, the required estimate will follow by an application of the Cauchy-Schwarz inequality, provided we can show that

$$\begin{aligned} C_1(m) := \int _{|{\varvec{p}}| \le 1} d{\varvec{p}}|{\varvec{p}}| \int _{\mathbb {R}}d{\varvec{q}}\,\omega _m({\varvec{q}})|\psi ({\varvec{p}},{\varvec{q}}) |^2 < +\infty . \end{aligned}$$
(A.5)

To this end, we write \(\psi ({\varvec{p}},{\varvec{q}}) = \hat{\varphi }'(-{\varvec{q}}) + \int _0^{\varvec{p}}d{\varvec{s}}\, \hat{\varphi }''({\varvec{s}}-{\varvec{q}})({\varvec{p}}-{\varvec{s}})\) and we recall that \(\hat{\varphi }\) is a Schwartz function, which entails the existence of a constant \(K > 0\) such that \(|\hat{\varphi }''({\varvec{s}})| \le K/(1+|{\varvec{s}}|)^2\). Using then the fact that

$$\begin{aligned} \int _{-{\varvec{q}}}^{{\varvec{p}}-{\varvec{q}}} \frac{d{\varvec{s}}}{(1+|{\varvec{s}}|)^2} = \frac{1}{1+|{\varvec{q}}|}-\frac{1}{1+|{\varvec{p}}-{\varvec{q}}|} \end{aligned}$$

for \(|{\varvec{q}}| > 1\), \(|{\varvec{p}}| \le 1\), we obtain the estimate

$$\begin{aligned}&\int _{|{\varvec{p}}| \le 1} d{\varvec{p}}|{\varvec{p}}| \int _{\mathbb {R}}d{\varvec{q}}\,\omega _m({\varvec{q}})\left| \int _0^{\varvec{p}}d{\varvec{s}}\, \hat{\varphi }''({\varvec{s}}-{\varvec{q}})({\varvec{p}}-{\varvec{s}})\right| ^2 \\&\quad \le 4K^2 \int _{|{\varvec{p}}| \le 1} d{\varvec{p}}|{\varvec{p}}|^3 \left[ |{\varvec{p}}|^2\int _{|{\varvec{q}}| \le 1} d{\varvec{q}}\,\omega _m({\varvec{q}}) \right. \\&\qquad \left. + \int _{|{\varvec{q}}| > 1} d{\varvec{q}}\,\omega _m({\varvec{q}})\left( \frac{1}{1+|{\varvec{q}}|}-\frac{1}{1+|{\varvec{p}}-{\varvec{q}}|}\right) ^2\right] < +\infty . \end{aligned}$$

This, together with the obvious estimate \(\int _{|{\varvec{p}}| \le 1} d{\varvec{p}}|{\varvec{p}}| \int _{\mathbb {R}}d{\varvec{q}}\,\omega _m({\varvec{q}})|\hat{\varphi }'(-{\varvec{q}})|^2 < +\infty \), shows, by the Minkowski inequality, the validity of (A.5). All in all, we obtain \(\Vert f\Vert _{0,-1} \le C \Vert f\Vert _{m,-1}\) with \(C = [\sqrt{1+m^2}+C_1(m)]^{1/2}\). \(\quad \square \)

We stress that, thanks to a general result by Buchholz [11, Appendix B], the condition 2. above is satisfied if \(\tilde{S}_m - \tilde{S}_0\) is a trace-class operator on \({\bar{K}}\). In order to get a more explicit expression for these operators, we observe that, thanks to (A.1) and [4, Lem. 3.2], there holds, for all \(m \ge 0\), the estimate

$$\begin{aligned} |{{\mathrm {Im}}\,}\langle f,g\rangle |^2 \le 4 \Vert f\Vert ^2_m \Vert g\Vert ^2_m, \qquad f,g \in \mathcal{D}_0(I), \end{aligned}$$

and therefore the symplectic form \({{\mathrm {Im}}\,}\langle \cdot ,\cdot \rangle \) extends by continuity to the real Hilbert space \((\overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}, {{\mathrm {Re}}\,}\langle \cdot ,\cdot \rangle _m)\), and there exist bounded antisymmetric operators \(R_m : \overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}\rightarrow \overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}\), \(m \ge 0\), such that

$$\begin{aligned} \frac{1}{2} {{\mathrm {Im}}\,}\langle f,g\rangle = {{\mathrm {Re}}\,}\langle f,R_m g\rangle _m, \qquad f,g \in \overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}. \end{aligned}$$

It is then a straigthforward computation to verify that (A.3) is satisfied if

$$\begin{aligned} \tilde{S}_m = \frac{1}{2} \left( \begin{array}{cc} \mathbf{1}&{} -R_m \\ R_m &{} \mathbf{1}\end{array}\right) , \quad \tilde{S}_0 = \frac{1}{2} \left( \begin{array}{cc} T &{} -R_m \\ R_m &{} T\end{array}\right) , \end{aligned}$$

with \(T : \overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}\rightarrow \overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}\) the (symmetric) bounded (by Proposition A.2) operator defined by

$$\begin{aligned} {{\mathrm {Re}}\,}\langle f, Tg\rangle _m = {{\mathrm {Re}}\,}\langle f,g\rangle _0, \qquad f,g \in \overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}. \end{aligned}$$

Therefore the condition that \(\tilde{S}_m - \tilde{S}_0\) is trace class is equivalent to the condition that \(\mathbf{1}- T\) is trace class on \(\overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m}\).

We also observe, in passing, that by [41, Lem. 3.3] the factoriality of \(\mathcal{C}^{(m)}(O_I)\) is equivalent to the fact that \(R_m\) or \(R_0\) (and then both) is injective with dense range. (Moreover, by antisymmetry, \(R_m\) is injective if and only if it has dense range.) If this is the case, it is also easy to check that \(T = R_m R_0^{-1}\).

We also put on record the following formula for the matrix elements of \(\mathbf{1}- T\):

$$\begin{aligned} {{\mathrm {Re}}\,}\langle f, (\mathbf {1}-T)g\rangle _m&= {{\mathrm {Re}}\,}\langle f,g\rangle _m - {{\mathrm {Re}}\,}\langle f,g\rangle _0 \nonumber \\&= \frac{1}{2}\int d{\varvec{p}}\,\left[ \left( \frac{1}{\omega _m({\varvec{p}})}-\frac{1}{|{\varvec{p}}|}\right) \overline{({{\mathrm {Re}}\,}f)^{\wedge }({\varvec{p}})}({{\mathrm {Re}}\,}g)^{\wedge }({\varvec{p}}) \right. \nonumber \\&\qquad \left. + \left( \omega _m({\varvec{p}})-|{\varvec{p}}|\right) \overline{({{\mathrm {Im}}\,}f)^{\wedge }({\varvec{p}})}({{\mathrm {Im}}\,}g)^{\wedge }({\varvec{p}})\right] . \end{aligned}$$
(A.6)

Thus we see that if we define \(\mathcal{H}_m^\pm (I) := \overline{\mathcal{D}_0(I,{\mathbb {R}})}^{\Vert \cdot \Vert _{m,\pm 1}}\), there holds \(\overline{\mathcal{D}_0(I)}^{\Vert \cdot \Vert _m} \cong \mathcal{H}_m^-(I) \oplus \mathcal{H}_m^+(I)\), and in this decomposition

$$\begin{aligned} \mathbf{1}- T = \left( \begin{matrix} Q^- &{}0 \\ 0 &{}Q^+ \end{matrix}\right) \end{aligned}$$

with bounded \(Q^\pm : \mathcal{H}_m^\pm (I) \rightarrow \mathcal{H}_m^\pm (I)\).

Lemma A.3

There holds

$$\begin{aligned} \langle f,Q^-g\rangle _{m,-1} = \int _{I \times I} d{\varvec{x}}d{\varvec{y}}\,f({\varvec{x}})g({\varvec{y}}) Q^-({\varvec{x}}-{\varvec{y}}), \qquad f,g \in \mathcal{D}_0(I,{\mathbb {R}}), \end{aligned}$$

with

$$\begin{aligned} Q^-({\varvec{x}}) := K_0(m|{\varvec{x}}|) + \log |{\varvec{x}}|, \qquad {\varvec{x}}\in {\mathbb {R}}, \end{aligned}$$

where \(K_0\) is the modified Bessel function of order zero. Moreover \(Q^-\) is continuously differentiable and with second derivative in \(L^2_{loc}({\mathbb {R}})\).

Proof

From (A.6) we see that \(\langle f,Q^-g\rangle _{m,-1} \) is the difference between the scalar field two-point functions for masses \(m>0\) and zero evaluated on the time zero line, and from, e.g., [37, 9.6.21], we obtain

$$\begin{aligned} \begin{aligned} \langle f,Q^-g\rangle _{m,-1}&= \frac{1}{2}\int _{\mathbb {R}}d{\varvec{p}}\,\left( \frac{1}{\omega _m({\varvec{p}})}-\frac{1}{|{\varvec{p}}|}\right) \overline{{\hat{f}}({\varvec{p}})}\hat{g}({\varvec{p}}) \\&= \int _{I \times I} d{\varvec{x}}d{\varvec{y}}\,f({\varvec{x}}) g({\varvec{y}})K_0(m|{\varvec{x}}-{\varvec{y}}|) - \frac{1}{2}\int _{\mathbb {R}}d{\varvec{p}}\,\frac{1}{|{\varvec{p}}|}\overline{{\hat{f}}({\varvec{p}})}\hat{g}({\varvec{p}}). \end{aligned}\end{aligned}$$

In order to treat the second term, we define functions \(F, G \in \mathcal{D}(I,{\mathbb {R}})\) by \(F({\varvec{x}}) := \int _{\infty }^{\varvec{x}}d{\varvec{y}}\,f({\varvec{y}})\), \(G({\varvec{x}}) := \int _{-\infty }^{\varvec{x}}d{\varvec{y}}\,g({\varvec{y}})\), so that \({\hat{f}}({\varvec{p}}) = -i{\varvec{p}}{\hat{F}}({\varvec{p}})\), \({\hat{g}}({\varvec{p}}) = -i{\varvec{p}}{\hat{G}}({\varvec{p}})\). With this definition, we get

$$\begin{aligned}\begin{aligned}&\int _{\mathbb {R}}d{\varvec{p}}\,\frac{1}{|{\varvec{p}}|}\overline{{\hat{f}}({\varvec{p}})}{\hat{g}}({\varvec{p}})\\&\quad = \int d{\varvec{p}}\,|{\varvec{p}}|\overline{{\hat{F}}({\varvec{p}})}{\hat{G}}({\varvec{p}}) \\&\quad = \lim _{\varepsilon \rightarrow 0^+} \int _{-\infty }^0 d{\varvec{p}}|{\varvec{p}}| e^{\varepsilon {\varvec{p}}}\overline{{\hat{F}}({\varvec{p}})}{\hat{G}}({\varvec{p}}) +\int _{0}^{+\infty } d{\varvec{p}}|{\varvec{p}}| e^{-\varepsilon {\varvec{p}}}\overline{{\hat{F}}({\varvec{p}})}{\hat{G}}({\varvec{p}}) \\&\quad = \lim _{\varepsilon \rightarrow 0^+} \int _{I \times I} d{\varvec{x}}d{\varvec{y}}F({\varvec{x}}) G({\varvec{y}})\left[ -\int _{-\infty }^0 d{\varvec{p}}\,{\varvec{p}}e^{i{\varvec{p}}({\varvec{y}}-{\varvec{x}}-i\varepsilon )}+\int _{0}^{+\infty } d{\varvec{p}}\, {\varvec{p}}e^{i{\varvec{p}}({\varvec{y}}-{\varvec{x}}+i\varepsilon )}\right] \\&\quad =- \lim _{\varepsilon \rightarrow 0^+} \int _{I \times I} d{\varvec{x}}d{\varvec{y}}F({\varvec{x}}) G({\varvec{y}})\left[ \frac{1}{({\varvec{y}}-{\varvec{x}}-i\varepsilon )^2}+\frac{1}{({\varvec{y}}-{\varvec{x}}+i\varepsilon )^2}\right] \\&\quad = -\lim _{\varepsilon \rightarrow 0^+} \int _{I \times I} d{\varvec{x}}d{\varvec{y}}F({\varvec{x}}) G({\varvec{y}})\frac{\partial ^2}{\partial {\varvec{x}}\partial {\varvec{y}}}\left[ \log [({\varvec{x}}-{\varvec{y}})^2+\varepsilon ^2]\right] \\&\quad = -\lim _{\varepsilon \rightarrow 0^+} \int _{I \times I} d{\varvec{x}}d{\varvec{y}}f({\varvec{x}}) g({\varvec{y}})\log [({\varvec{x}}-{\varvec{y}})^2+\varepsilon ^2]\\&\quad = -2\int _{I \times I} d{\varvec{x}}d{\varvec{y}}f({\varvec{x}}) g({\varvec{y}})\log |{\varvec{x}}-{\varvec{y}}|, \end{aligned}\end{aligned}$$

where the last equality is obtained by the dominated convergence theorem, observing that for \(({\varvec{x}}- {\varvec{y}})^2 +\varepsilon ^2 < 1\) the inequality

$$\begin{aligned} |f({\varvec{x}}) g({\varvec{y}})\log [({\varvec{x}}-{\varvec{y}})^2+\varepsilon ^2]| \le |f({\varvec{x}}) g({\varvec{y}})\log [({\varvec{x}}-{\varvec{y}})^2]| \end{aligned}$$

holds, and the right hand side is an integrable function.

Finally, the last statement is obtained from the fact that there exist analytic functions \(\varphi _1\), \(\varphi _2\) defined in a neighbourhood of the origin, such that [37, 9.6.12-13]

$$\begin{aligned} Q^-({\varvec{x}}) = -m^2 {\varvec{x}}^2\log (m|{\varvec{x}}|)\varphi _1(m^2{\varvec{x}}^2)+\varphi _2(m^2{\varvec{x}}^2), \end{aligned}$$

and moreover \({\varvec{x}}\mapsto \log |{\varvec{x}}|\) belongs to \(L^2_{loc}({\mathbb {R}})\). \(\quad \square \)

Lemma A.4

The operator \(Q^-\) is of trace class on \(\mathcal{H}^-_m(I)\).

Proof

We can of course assume \(I =(-1,1)\). By the change of variables \({\varvec{\xi }}:= {\varvec{x}}-{\varvec{y}}\), \({\varvec{\eta }}:= {\varvec{x}}+{\varvec{y}}\) we can write

$$\begin{aligned} \langle f, Q^-g\rangle _{m,-1} = \frac{1}{2} \int _{-2}^2 d{\varvec{\xi }}\left[ \int _{-2+|{\varvec{\xi }}|}^{2-|{\varvec{\xi }}|} d{\varvec{\eta }}\, f\Big (\frac{{\varvec{\eta }}+{\varvec{\xi }}}{2}\Big ) g\Big (\frac{{\varvec{\eta }}-{\varvec{\xi }}}{2}\Big )\right] Q^-({\varvec{\xi }})\varphi ({\varvec{\xi }}), \end{aligned}$$

with \(\varphi \in \mathcal{D}({\mathbb {R}},{\mathbb {R}})\) such that \(\varphi ({\varvec{\xi }}) = 1\) for \(|{\varvec{\xi }}| \le 2\) and \(\varphi ({\varvec{\xi }}) = 0\) for \(|{\varvec{\xi }}| \ge 3\). We now expand the function \(Q^- \varphi \) in Fourier series on the interval \([-4, 4]\), obtaining

$$\begin{aligned} Q^-({\varvec{\xi }})\varphi ({\varvec{\xi }}) = \sum _{k \in {\mathbb {Z}}} Q_k^- e^{i\frac{\pi k}{4}{\varvec{\xi }}}, \end{aligned}$$

with a totally convergent expansion, that is \(\sum _{k \in {\mathbb {Z}}} |Q_k^-| < \infty \), since the periodized \(Q^- \varphi |_{[-4,4]}\) is in \(C^1({\mathbb {R}})\). Actually, \(Q_k^- = -\frac{{Q_k}''}{k^2}\), with \(Q''_k\) the Fourier coefficients of \(\frac{d^2}{d{\varvec{\xi }}^2}(Q^-\varphi ) \in L^2([-4,4])\) (by the previous Lemma), and therefore, by the Cauchy-Schwartz and Bessel inequalities,

$$\begin{aligned} \sum _{k \in {\mathbb {Z}}\setminus \{0\}} |Q_k^-| |k| = \sum _{k \in {\mathbb {Z}}\setminus \{0\}} \frac{|Q''_k|}{|k|} \le \bigg [\sum _{k \in {\mathbb {Z}}\setminus \{0\}} |Q''_k|^2\bigg ]^{\frac{1}{2}} \bigg [\sum _{k \in {\mathbb {Z}}\setminus \{0\}} \frac{1}{k^2}\bigg ]^{\frac{1}{2}} < +\infty . \end{aligned}$$

It is therefore legitimate to interchange the series with the \({\varvec{\xi }}\)-integration:

$$\begin{aligned} \langle f, Q^-g\rangle _{m,-1} = \sum _{k \in {\mathbb {Z}}} Q_k^- \int _{I} d{\varvec{x}}\, f({\varvec{x}}) e^{i\frac{\pi k}{4}{\varvec{x}}} \int _I d{\varvec{y}}\, g({\varvec{y}}) e^{-i\frac{\pi k}{4}{\varvec{y}}}. \end{aligned}$$

Let now \(\chi \in \mathcal{D}({\mathbb {R}},{\mathbb {R}})\) be such that \(\chi ({\varvec{x}}) = 1\) for \(|{\varvec{x}}| \le 1\) and \(\chi ({\varvec{x}}) = 0\) for \(|{\varvec{x}}| \ge 2\). Then

$$\begin{aligned}\begin{aligned} \int _{I} d{\varvec{x}}\, f({\varvec{x}}) e^{i\frac{\pi k}{4}{\varvec{x}}}&= \int _{I} d{\varvec{x}}\, f({\varvec{x}}) \chi ({\varvec{x}})\Big [\cos \Big (\frac{\pi k}{4}{\varvec{x}}\Big )+ i \sin \Big (\frac{\pi k}{4}{\varvec{x}}\Big )\Big ]\\&=\frac{1}{2\pi }\left[ \int _{{\mathbb {R}}} \frac{d{\varvec{p}}}{\omega _m({\varvec{p}})} \overline{{\hat{f}}({\varvec{p}})} {\hat{\psi }}_k^-({\varvec{p}})+i\int _{{\mathbb {R}}} \frac{d{\varvec{p}}}{\omega _m({\varvec{p}})} \overline{{\hat{f}}({\varvec{p}})} \hat{\varphi }_k^-({\varvec{p}})\right] , \end{aligned}\end{aligned}$$

with \({\hat{\psi }}_k^-({\varvec{p}}) := \frac{\omega _m({\varvec{p}})}{2}[ {\hat{\chi }}({\varvec{p}}+\frac{\pi k}{4})+{\hat{\chi }}({\varvec{p}}-\frac{\pi k}{4})]\), \({\hat{\varphi }}_k^-({\varvec{p}}) := \frac{\omega _m({\varvec{p}})}{2i}[ {\hat{\chi }}({\varvec{p}}+\frac{\pi k}{4})-{\hat{\chi }}({\varvec{p}}-\frac{\pi k}{4})]\). Extending the definitions of \(\Vert \cdot \Vert _{m,-1}\) and \(\langle \cdot , \cdot \rangle _{m,-1}\) to \(\mathcal{S}({\mathbb {R}},{\mathbb {R}})\), and observing that, for all \(k \in {\mathbb {Z}}\),

$$\begin{aligned}\begin{aligned} \int _{\mathbb {R}}d{\varvec{p}}\, \omega _m({\varvec{p}}) \Big |{\hat{\chi }}\Big ({\varvec{p}}+\frac{\pi k}{4}\Big )\Big |^2&= \int _{\mathbb {R}}d{\varvec{p}}\, \omega _m\Big ({\varvec{p}}-\frac{\pi k}{4}\Big ) |{\hat{\chi }} ({\varvec{p}})|^2\\&\le C \int _{\mathbb {R}}d{\varvec{p}}\, \Big (1+\Big |{\varvec{p}}-\frac{\pi k}{4}\Big |\Big ) |{\hat{\chi }} ({\varvec{p}})|^2 \le C_1 + C_2 |k|, \end{aligned}\end{aligned}$$

for suitable constants \(C_1, C_2 > 0\), one gets the estimates \(\Vert \psi _k^-\Vert _{m,-1}, \Vert \varphi _k^-\Vert _{m,-1} \le \sqrt{C_1+C_2|k|}\). Thus in particular \(\psi _k^-,\varphi _k^- \in \overline{\mathcal{S}({\mathbb {R}},{\mathbb {R}})}^{\Vert \cdot \Vert _{m,-1}}\) for all \(k \in {\mathbb {Z}}\), and therefore, if \(P^- : \overline{\mathcal{S}({\mathbb {R}},{\mathbb {R}})}^{\Vert \cdot \Vert _{m,-1}} \rightarrow \mathcal{H}_m^-(I)\) denotes the orthogonal projection,

$$\begin{aligned} \int _{I} d{\varvec{x}}\, f({\varvec{x}}) e^{i\frac{\pi k}{4}{\varvec{x}}}= & {} \frac{1}{2\pi }[\langle f, \psi _k^-\rangle _{m,-1}+i\langle f,\varphi _k^-\rangle _{m,-1}] \\= & {} \frac{1}{2\pi } [\langle f, P^-\psi _k^-\rangle _{m,-1}+i\langle f, P^-\varphi _k^-\rangle _{m,-1}], \end{aligned}$$

and, taking into account the fact that \(\langle f, Q^- g\rangle _{m,-1}\) is real,

$$\begin{aligned} \begin{aligned} \langle f, Q^- g\rangle _{m,-1} =&\frac{1}{4\pi ^2} \sum _{k \in {\mathbb {Z}}} \big [{{\mathrm {Re}}\,}Q_k^-\big ( \langle f,P^-\psi _k^-\rangle _{m,-1} \langle P^-\psi _k^-,g\rangle _{m,-1} \\&+\langle f,P^-\varphi _k^-\rangle _{m,-1} \langle P^-\varphi _k^-,g\rangle _{m,-1}\big )\\&-{{\mathrm {Im}}\,}Q_k^-\big ( \langle f,P^-\varphi _k^-\rangle _{m,-1} \langle P^-\psi _k^-,g\rangle _{m,-1}\\&+\langle f,P^-\psi _k^-\rangle _{m,-1} \langle P^-\varphi _k^-,g\rangle _{m,-1}\big )\big ]. \end{aligned}\end{aligned}$$

Now, since by the above estimates

$$\begin{aligned} \sum _{k \in {\mathbb {Z}}} |Q_k^-| \Vert P^-\psi _k^-\Vert _{m,-1}^2 \le C_1 \sum _{k \in {\mathbb {Z}}} |Q_k^-|+C_2\sum _{k \in {\mathbb {Z}}} |Q_k^-| |k| < +\infty , \end{aligned}$$

and similarly for \(\sum _{k \in {\mathbb {Z}}} |Q_k^-| \Vert P^-\varphi _k^-\Vert _{m,-1}^2\), \(\sum _{k \in {\mathbb {Z}}} |Q_k^-| \Vert P^-\psi _k^-\Vert _{m,-1}\Vert P^-\varphi _k^-\Vert _{m,-1}\), and recalling that \(\mathcal{D}_0(I,{\mathbb {R}})\) is dense in \(\mathcal{H}_m^-(I)\), we obtain the thesis using [42, Thm. 7.12]. \(\square \)

Lemma A.5

There holds

$$\begin{aligned} \langle f,Q^+g\rangle _{m,1} = \int _{I \times I} d{\varvec{x}}d{\varvec{y}}\,f({\varvec{x}})g({\varvec{y}}) Q^+({\varvec{x}}-{\varvec{y}}), \qquad f,g \in \mathcal{D}_0(I,{\mathbb {R}}), \end{aligned}$$

with

$$\begin{aligned} Q^+({\varvec{x}}) := -\frac{m}{|{\varvec{x}}|}K_1(m|{\varvec{x}}|) + \frac{1}{\, |{\varvec{x}}|^2}, \qquad {\varvec{x}}\in {\mathbb {R}}, \end{aligned}$$

where \(K_1\) is the modified Bessel function of order one. Moreover \(Q^+\) is in \(L^2_{loc}({\mathbb {R}})\).

Proof

We start observing that, by differentiating under the integral sign, one gets

$$\begin{aligned} \frac{\partial }{\partial m} \int _{\mathbb {R}}d{\varvec{p}}\, {\omega _m({\varvec{p}})} \overline{{\hat{f}}({\varvec{p}})}{\hat{g}}({\varvec{p}}) = m \int _{\mathbb {R}}\,\frac{d{\varvec{p}}}{\omega _m({\varvec{p}})}\overline{{\hat{f}}({\varvec{p}})}{\hat{g}}({\varvec{p}}) \ , \quad m > 0 \ . \end{aligned}$$

Since the r.h.s. is integrable in a right neighbourhood of \(m=0\),

$$\begin{aligned} \langle f,Q^+g\rangle _{m,1}&= \frac{1}{2} \int _{\mathbb {R}}d{\varvec{p}}\, ({\omega _m({\varvec{p}})}-|{\varvec{p}}|) \overline{{\hat{f}}({\varvec{p}})}{\hat{g}}({\varvec{p}}) \\&= \frac{1}{2} \int _0^m dm' m' \int _{\mathbb {R}}\,\frac{d{\varvec{p}}}{\omega _{m'}({\varvec{p}})}\overline{{\hat{f}}({\varvec{p}})}{\hat{g}}({\varvec{p}}) \\&= \int _0^m dm' m' \int _{I \times I} d{\varvec{x}}d{\varvec{y}}\,f({\varvec{x}}) g({\varvec{y}})K_0(m'|{\varvec{x}}-{\varvec{y}}|) \ . \end{aligned}$$

By [37, 9.6.13], we have \(|K_0(z)| \le C_1 |\log z| + C_2\), \(|z| \le 2m\), for suitable constants \(C_1\) and \(C_2\) and, since

$$\begin{aligned}\int _0^m dm' m' |\log (m'|{\varvec{x}}-{\varvec{y}}|)|= {\left\{ \begin{array}{ll} \frac{1}{2|{\varvec{x}}- {\varvec{y}}|^2} + \frac{m^2}{2} \left( \log (m|{\varvec{x}}-{\varvec{y}}|) - \frac{1}{2}\right) &{} |{\varvec{x}}- {\varvec{y}}| \ge \frac{1}{m} \\ - \frac{m^2}{2} \left( \log (m|{\varvec{x}}-{\varvec{y}}|) - \frac{1}{2}\right) &{} |{\varvec{x}}- {\varvec{y}}| \le \frac{1}{m} \end{array}\right. }\end{aligned}$$

we can apply Fubini’s Theorem and interchange the order of the integrals, thus obtaining

$$\begin{aligned}\langle f,Q^+g\rangle _{m,1} = \int _{I \times I} d{\varvec{x}}d{\varvec{y}}\,f({\varvec{x}}) g({\varvec{y}}) \int _0^m dm' m' K_0(m'|{\varvec{x}}-{\varvec{y}}|) \ . \end{aligned}$$

Making use of [37, 9.6.28], we see that \(m' K_0(m'|{\varvec{x}}-{\varvec{y}}|) = \frac{1}{|{\varvec{x}}- {\varvec{y}}|^2} \frac{\partial }{\partial m'} \big (-m' |{\varvec{x}}- {\varvec{y}}| K_1(m' |{\varvec{x}}- {\varvec{y}}|) \big )\) and therefore, also observing that \(\lim _{z \rightarrow 0} z K_1(z) = 1\) by [37, 9.6.11], we find the required expression for \(Q^+\) as in the statement.

Finally, the last claim readily follows again from [37, 9.6.11] according to which

$$\begin{aligned}Q^+({\varvec{x}}) = \varphi _1(m^2 {\varvec{x}}^2) + \log (m |{\varvec{x}}|)\varphi _2(m^2 {\varvec{x}}^2)\end{aligned}$$

for suitable analytic functions \(\varphi _1\) and \(\varphi _2\). \(\quad \square \)

Lemma A.6

The operator \(Q^+\) is of trace class on \(\mathcal{H}^+_m(I)\).

Proof

As in the proof of Lemma A.4, we assume that \(I=(-1,1)\). We also employ \(\varphi \) and \(\chi \) as defined there. We define \(F({\varvec{\xi }}) = \int _{-2+|{\varvec{\xi }}|}^{2-|{\varvec{\xi }}|} d{\varvec{\eta }}\, f\Big (\frac{{\varvec{\eta }}+{\varvec{\xi }}}{2}\Big ) g\Big (\frac{{\varvec{\eta }}-{\varvec{\xi }}}{2}\Big )\) and \(R({\varvec{\xi }}) = \int _{-2}^{\varvec{\xi }}d{\varvec{\xi }}' Q^+({\varvec{\xi }}').\) Therefore,

$$\begin{aligned} \langle f, Q^+g\rangle _{m,1} = \frac{1}{2} \int _{-2}^2 d{\varvec{\xi }}F({\varvec{\xi }}) Q^+({\varvec{\xi }}) = - \frac{1}{2} \int _{-2}^2 d{\varvec{\xi }}F'({\varvec{\xi }})R({\varvec{\xi }}) \varphi ({\varvec{\xi }}) \end{aligned}$$
(A.7)

where we used integration by parts, the fact that \(F(\pm 2)=0\) and that \(F'({\varvec{\xi }})\) exists for all \({\varvec{\xi }}\ne 0\) and is integrable in \([-2,2]\). We expand the function \(R \varphi \) in Fourier series on the interval \([-4, 4]\), obtaining

$$\begin{aligned} R({\varvec{\xi }})\varphi ({\varvec{\xi }}) = \sum _{k \in {\mathbb {Z}}} R_k e^{i\frac{\pi k}{4}{\varvec{\xi }}}, \end{aligned}$$

and notice that, again integrating by parts, \(R_k = \frac{1}{2\pi i k} \int _{-4}^4 d{\varvec{\xi }}(R\varphi )' ({\varvec{\xi }}) e^{-i\frac{\pi k}{4}{\varvec{\xi }}} =: \frac{ 4 Q^+_k}{i \pi k}\), \(k \ne 0\). Since \((R\varphi )' \in L^2([-4,4])\) by the previous lemma, it follows that \(\sum _k |Q^+_k|^2 < +\infty \) and hence

$$\begin{aligned}\sum _{k \ne 0} |R_k| = \frac{4}{\pi }\sum _{k \ne 0} \frac{|Q^+_k|}{|k|} \le \frac{4}{\pi }\left( \sum _{k \ne 0} |Q^+_k|^2\right) ^{1/2} \left( \sum _{k \ne 0} \frac{1}{k^2} \right) ^{1/2} < +\infty \ . \end{aligned}$$

We are then allowed to interchange the integral with the series in (A.7) and obtain

$$\begin{aligned} \langle f, Q^+g\rangle _{m,1}&= - \frac{1}{2} \sum _k R_k \int _{-2}^2 d{\varvec{\xi }}F'({\varvec{\xi }}) e^{i\frac{\pi k}{4}{\varvec{\xi }}} = \frac{1}{2} \sum _k R_k \frac{i \pi k}{4} \int _{-2}^2 d{\varvec{\xi }}F({\varvec{\xi }}) e^{i\frac{\pi k}{4}{\varvec{\xi }}} \\&= \sum _{k\ne 0} Q^+_k \int _{I} d{\varvec{x}}\, f({\varvec{x}}) e^{i\frac{\pi k}{4}{\varvec{x}}} \int _I d{\varvec{y}}\, g({\varvec{y}}) e^{-i\frac{\pi k}{4}{\varvec{y}}}. \end{aligned}$$

Now, similarly to the proof of Lemma A.4,

$$\begin{aligned} \int _{I} d{\varvec{x}}\, f({\varvec{x}}) e^{i\frac{\pi k}{4}{\varvec{x}}} =\frac{1}{2\pi }\left[ \int _{{\mathbb {R}}} d{\varvec{p}}\, \omega _m({\varvec{p}}) \overline{{\hat{f}}({\varvec{p}})} {\hat{\psi }}_k^+({\varvec{p}})+i\int _{{\mathbb {R}}} d{\varvec{p}}\, \omega _m({\varvec{p}}) \overline{{\hat{f}}({\varvec{p}})} {\hat{\varphi }}_k^+({\varvec{p}})\right] , \end{aligned}$$

with \({\hat{\psi }}_k^+({\varvec{p}}) := \frac{1}{2 \omega _m({\varvec{p}})}[ {\hat{\chi }}({\varvec{p}}+\frac{\pi k}{4})+{\hat{\chi }}({\varvec{p}}-\frac{\pi k}{4})]\), \(\hat{\varphi }_k^+({\varvec{p}}) := \frac{1}{2i \omega _m({\varvec{p}})}[ {\hat{\chi }}({\varvec{p}}+\frac{\pi k}{4})-{\hat{\chi }}({\varvec{p}}-\frac{\pi k}{4})]\). Observe that, for all \({\varvec{p}}\in {\mathbb {R}}\),

$$\begin{aligned}\sup _{{\varvec{q}}\in {\mathbb {R}}} \frac{|{\varvec{q}}|}{\omega _m({\varvec{p}}-{\varvec{q}})} = \sup _{{\varvec{q}}\in {\mathbb {R}}} \frac{|{\varvec{p}}- {\varvec{q}}|}{\omega _m({\varvec{q}})} \le C(1+|{\varvec{p}}|)\end{aligned}$$

for a suitable \(C>0\). Therefore, for all \(k \in {\mathbb {Z}}\setminus \{0\}\),

$$\begin{aligned} \int _{\mathbb {R}}d{\varvec{p}}\frac{1}{\omega _m({\varvec{p}})} \Big |{\hat{\chi }}\Big ({\varvec{p}}+\frac{\pi k}{4}\Big )\Big |^2= & {} \int _{\mathbb {R}}\frac{d{\varvec{p}}}{\omega _m\Big ({\varvec{p}}-\frac{\pi k}{4}\Big )} |{\hat{\chi }} ({\varvec{p}})|^2 \\\le & {} \frac{4 C}{\pi |k| } \int _{\mathbb {R}}d{\varvec{p}}(1 + |{\varvec{p}}|) |{\hat{\chi }} ({\varvec{p}})|^2 \ , \end{aligned}$$

which implies that \(\Vert \psi _k^+\Vert _{m,1}, \Vert \varphi _k^+\Vert _{m,1} \le \frac{C'}{\sqrt{ |k|}}\), \(k \ne 0\). Thus, we obtain that \(\sum _k |Q_k^+| \, \Vert P^+ \psi _k^+\Vert _{m,1}^2\), \(\sum _k |Q_k^+| \, \Vert P^+ \varphi _k^+\Vert _{m,1}^2\) and \(\sum _k |Q_k^+| \, \Vert P^+ \psi _k^+\Vert _{m,1} \Vert P^+ \varphi _k^+\Vert _{m,1}\) are all convergent, and the conclusion follows as in the proof of Lemma A.4. \(\quad \square \)

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Conti, R., Morsella, G. Asymptotic Morphisms and Superselection Theory in the Scaling Limit II: Analysis of Some Models. Commun. Math. Phys. 376, 1767–1801 (2020). https://doi.org/10.1007/s00220-019-03564-8

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