Compton scattering in the Buchholz–Roberts framework of relativistic QED

Abstract

We consider a Haag–Kastler net in a positive energy representation, admitting massive Wigner particles and asymptotic fields of massless bosons. We show that massive single-particle states are always vacua of the massless asymptotic fields. Our argument is based on the Mean Ergodic Theorem in a certain extended Hilbert space. As an application of this result, we construct the outgoing isometric wave operator for Compton scattering in QED in a class of representations recently proposed by Buchholz and Roberts. In the course of this analysis, we use our new technique to further simplify scattering theory of massless bosons in the vacuum sector. A general discussion of the status of the infrared problem in the setting of Buchholz and Roberts is given.

Appendices

Appendix A: Mean Ergodic Theorem and invariant vectors

We pick h as in (3.7) and recall a variant of the abstract Mean Ergodic Theorem:

Theorem A.1

Let S be a self-adjoint operator on (a domain in) $\mathcal {H}$ and $F_S$ its spectral measure. Then,

$$\begin{aligned} {{\mathrm{s-lim}}}_{t\rightarrow \infty }\int \mathrm{d}t'\,h_t(t')\mathrm {e}^{\mathrm {i}t'S}=F_S(\{0\}). \end{aligned}$$

(A.1)

Now, we determine the projection $F_S(\{0\})$ on the subspace of invariant vectors of $t\mapsto \mathrm {e}^{\mathrm {i}tS}$ for the relevant operators S.

Proposition A.2

Let $(H,\varvec{P})$ be the energy–momentum operators of a Haag–Kastler theory and E their joint spectral measure.

• (a) Let $S_{\nu }:= H-\cos \,\nu |\varvec{P}|$ and $F_{S_{\nu }}$ be the spectral measure of $S_{\nu }$. Then,

  $$\begin{aligned} F_{S_{\nu }}(\{0\})&= \left\{ \begin{array}{ll} E(\partial \overline{V}_+) &{}\quad \text {for}\,\, \nu =0, \\ E(\{0\}) &{} \quad \text {for}\,\, \nu \in (0,\pi ]. \\ \end{array} \right. \end{aligned}$$

  (A.2)

• (b) Let $S_{\nu ,\varvec{\xi }}:=H-|\varvec{\xi }|\cos \,\nu -\omega _m(\varvec{P}+\varvec{\xi })$, where $\omega _m(\varvec{p})=\sqrt{\varvec{p}^2+m^2}$, and $F_{S_{\nu ,\varvec{\xi }}}$ be the spectral measure of $S_{\nu ,\varvec{\xi }}$. Then, for $\varvec{\xi }\ne 0$,

  $$\begin{aligned} F_{ S_{\nu ,\varvec{\xi }} }(\{0\})&= \left\{ \begin{array}{ll} 0 &{}\quad \text {for}\,\, \nu \in [0,\pi ){ or}m>0, \\ E(\{0\}) &{}\quad \text {for}\,\, \nu =\pi \,\text {and}\, m=0. \end{array} \right. \end{aligned}$$

  (A.3)

Proof

(a) For $\Psi _{0}\in \mathrm {Ran}F_{S_0}(\{0\})$, we have $(H-|\varvec{P}|)\Psi _0=0$; hence, $\Psi _0\in \mathrm {Ran}E(\partial \overline{V}_+)$. This gives the first part of (A.2). To check the second part, we note that for $\nu \in (0,\pi ]$ the set

$$\begin{aligned} \Delta _{\nu }:=\{\, (p^0, \varvec{p})\,|\, p^0=\cos \nu \,|\varvec{p}| \,\} \end{aligned}$$

(A.4)

intersects with $\overline{V}_+$ only at $\{0\}$.

(b) First, we note that the set

$$\begin{aligned} \Delta _{\nu ,\varvec{\xi }}:=\{\, (p^0, \varvec{p})\,|\, p^0=|\varvec{\xi }|\cos \nu +\omega _m(\varvec{p}+\varvec{\xi }) \,\} \end{aligned}$$

(A.5)

describes a mass hyperboloid shifted by a spacelike or lightlike vector $(|\varvec{\xi }|\cos \nu ,-\varvec{\xi })$. Thus, $\Delta _{\nu ,\varvec{\xi }}$ contains zero only if $m=0$ and $\nu =\pi$. Hence, it suffices to show that the relation

$$\begin{aligned} (H-\omega _m(\varvec{P}+\varvec{\xi }))\Psi =|\varvec{\xi }|\cos \nu \Psi , \end{aligned}$$

(A.6)

where $\Psi =E(\Delta )\Psi$, $\Delta$ compact, can only hold for $\Psi \in E(\{0\})\mathcal {H}$. This is shown by generalizing an argument from Appendix of [4]. $\square$

Appendix B: Admissible propagation observables

Definition B.1

Let $[1,\infty )\ni t\mapsto A_t\in B(\mathcal {H})$ be a propagation observable, H a positive, self-adjoint operator on a domain D(H) in $\mathcal {H}$, and $D,D^*\subset \mathcal {H}$ some dense domains. We say that A is admissible if:

• (a) For any $\Psi \in D^{(*)},$ the limit $\lim _{t\rightarrow \infty }A_t^{(*)}\Psi$ exists.

• (b) $\sup _{t\in [1,\infty )}\Vert A_t^{(*)}(1+H)^{-1}\Vert <\infty$.

• (c) Set $A_t(s):=\mathrm {e}^{\mathrm {i}sH}A_t\mathrm {e}^{-\mathrm {i}sH}$. All the derivatives $A_t^{(n)}=\partial ^n_s A_t(s)|_{s=0}$ exist in norm and satisfy (a), (b).

Here, $(*)$ means that the statement holds with and without all $*$ symbols (correlated).

As shown in the next two propositions, limits of admissible propagation observables exist as closable operators on the following dense domain:

$$\begin{aligned} D_H:=\bigcap _{n\ge 1} D(H^n). \end{aligned}$$

(B.1)

Moreover, $D_H$ is an invariant domain of these limits.

Proposition B.2

Let A be an admissible propagation observable. Then:

• (a) For any $\Psi \in D_H,$ the limit $\lim _{t\rightarrow \infty }A_t\Psi$ exists and defines a closable operator $A^{\mathrm {out}}$ on $D_H$. This operator is uniquely specified by its values on D.

• (b) $A^{\mathrm {out}} D_H\subset D_H$.

Proof

Exploiting part (c) of Definition B.1, we write

$$\begin{aligned} A_t\Psi =(1+H)^{-1}(-\mathrm {i})A^{(1)}_t\Psi +(1+H)^{-1}A_t(1+H)\Psi . \end{aligned}$$

(B.2)

Vectors $\Psi ,(1+H)\Psi$ appearing on the right-hand side of (B.2) can be approximated uniformly in t by elements of D (cf. Definition B.1 (b), (c)). By part (a) of Definition B.1, $A_t, A^{(1)}_t$ converge on D which gives the existence of $A^{\mathrm {out}}$ as an operator on $D_H$. Since the above reasoning applies also to $A_t^*$, the operator $A^{\mathrm {out}}$ is closable. To show that it is uniquely determined by its values on D, consider admissible propagation observables $A_1$ and $A_2$ such that $\lim _{t\rightarrow \infty }A_{1,t}\Phi =\lim _{t\rightarrow \infty }A_{2,t}\Phi$ for $\Phi \in D$. Then, it is clear from the above discussion that $A_1^{\mathrm {out}}=A_2^{\mathrm {out}}$ as operators on $D_H$. This completes the proof of (a).

To prove (b), we make use of a standard commutator formula (see e.g. [15])

$$\begin{aligned}{}[(1+H)^\ell ,A_t]&=\sum _{k=1}^{\ell }\begin{pmatrix} \ell \\ k \end{pmatrix} \text {ad}_{H}^k(A_t)(1+H)^{\ell -k},\end{aligned}$$

(B.3)

$$\begin{aligned} \text {ad}_H^0(A_t):=A_t,&\qquad \text {ad}_H^k(A_t):=[H,\text {ad}_H^{k-1}(A_t)], \end{aligned}$$

(B.4)

which holds as an equality of quadratic forms on $D_H\times D_H$. Exploiting part (c) of Definition B.1, which ensures that $\text {ad}_{H}^k(A_t)=(-\mathrm {i})^kA^{(k)}_t$ are bounded operators, we obtain for any $\Psi \in D_H$

$$\begin{aligned} A_t\Psi =(1+H)^{-\ell }\left( \sum _{k=0}^{\ell }\begin{pmatrix} \ell \\ k \end{pmatrix} (-\mathrm {i})^kA^{(k)}_t (1+H)^{\ell -k}\right) \Psi , \end{aligned}$$

(B.5)

where we set by convention $A_t^{(0)}=A_t$. Taking now the limit $t\rightarrow \infty$ on both sides of (B.5), we obtain (b). $\square$

Proposition B.3

Let $A_i$, $i=1,\ldots , n$, be admissible propagation observables. Then, for any $\Psi \in D_H$,

$$\begin{aligned} A_1^{\mathrm {out}}\ldots A_n^{\mathrm {out}}\Psi =\lim _{t\rightarrow \infty }A_{1,t}\ldots A_{n,t}\Psi . \end{aligned}$$

(B.6)

Proof

For $n=1,$ the statement follows from Proposition B.2. We suppose now it holds for some $n>1$ and prove it for $n+1$. Similarly as in (B.2), $\Psi \in D_H,$

$$\begin{aligned} A_{1,t}\ldots A_{n+1,t}\Psi&=A_{1,t}(1+H)^{-1}(-\mathrm {i}) \sum _{\ell =2}^{n+1}A_{2,t}\ldots A^{(1)}_{\ell ,t} \ldots A_{n+1,t}\Psi \nonumber \\&\phantom {44}+A_{1,t}(1+H)^{-1}A_{2,t}\ldots A_{n+1,t}(1+H)\Psi . \end{aligned}$$

(B.7)

By the induction hypothesis and Proposition B.2 the above expression converges strongly as $t\rightarrow \infty$. Next, we pick $\Phi \in D_H$ and write

$$\begin{aligned} \langle \Phi , A_{1,t}\ldots A_{n+1,t}\Psi \rangle&=\langle \Phi , A_{1,t}A_{2}^{\mathrm {out}}\ldots A_{n+1}^{\mathrm {out}}\Psi \rangle +o(t^{0})\nonumber \\&=\langle \Phi , A_{1}^{\mathrm {out}}A_{2}^{\mathrm {out}}\ldots A_{n+1}^{\mathrm {out}}\Psi \rangle +o(t^{0}), \end{aligned}$$

(B.8)

where in the first step we used the induction hypothesis, in the second step Proposition B.2 and $o(t^0)$ denotes terms which tend to zero as $t\rightarrow \infty$. This concludes the proof. $\square$

Appendix C: Geometric argument

We refer to Sect. 2 and to [9, Appendix] for a brief summary of relevant geometric concepts.

Fig. 2

Geometrical situation in Lemma 5.2 for the case $\Lambda =I$. The double cone $\mathcal {O}$ is shifted into lightlike directions determined by $\Theta \subset S^2$. The resulting union of shifted double cones gives the region $\mathcal {U}$ in accordance with Eq. (C.1). As shown below, $\mathcal {U}$ is in $\mathcal {C}^{\mathrm {c}}$ which is indicated by the dotted lines

Proof of Lemma 5.2

We prove the statement only for $\Lambda =I$, since the generalization to small Lorentz transformations then easily follows. First, we note that

$$\begin{aligned} \bigcup _{t\ge 1} O_t= \bigcup _{t\ge 1} \bigcup \limits _{\tau \in t+t^{\bar{\varepsilon }}\mathrm {supp}h}\left\{ \mathcal {O}+\tau (1, \Theta )\right\} \subset \bigcup _{\tau \in \mathbb {R}_+} \left\{ \mathcal {O}+ \tau (1, \Theta )\right\} =:\mathcal {U}. \end{aligned}$$

(C.1)

We will show that for any double cone $\mathcal {O}\in \mathcal {K}$ and open $\Theta \subset S^2$ with $\overline{\Theta }\subsetneq S^2$, there is a future lightcone V and a hypercone $\mathcal {C}\subset \mathcal {F}_V$ such that the corresponding set $\mathcal {U}$ given by (C.1) is in $\mathcal {C}^{\mathrm {c}}$. Such a situation is depicted in Fig. 2.

We fix a future lightcone V so that $\overline{\mathcal {O}}\subset V$ and choose a coordinate frame in which the origin is at the apex of V. Next, we use the fact that there is an $\varvec{\ell } _0\in S^2$ and an $1\ge \varepsilon _0>0$ such that the spherical cap

$$\begin{aligned} \Theta _{\varepsilon }:=\{ \varvec{\ell } \in S^2\,|\, 1-\varepsilon \le \varvec{\ell } \varvec{\ell } _0\le 1\,\} \end{aligned}$$

(C.2)

is contained in $S^2\backslash \overline{\Theta }$ for all $0<\varepsilon \le \varepsilon _0$. Let, moreover, $\mathsf {K}_{\varepsilon }$ be a cone in the unit ball $\mathsf {B}$ with apex at $\varvec{u}_{\varepsilon }:=(1-\varepsilon )\varvec{\ell } _0$ and the opening angle determined by $\Theta _{\varepsilon }$. More precisely,

$$\begin{aligned} \mathsf {K}_{\varepsilon }:=\left\{ \varvec{u}\in \mathsf {B}\,|\, \varvec{u}=\varvec{u}_{\varepsilon }+s\,(\varvec{\ell } -\varvec{u}_{\varepsilon } ),\ 0\le s<1, \ \varvec{\ell } \in \Theta _{\varepsilon } \right\} . \end{aligned}$$

(C.3)

Using the Beltrami–Klein map $\varvec{v}: \mathsf {H}_{\bar{\tau }}\rightarrow \mathsf {B}$ given by $\varvec{v}(a)=\varvec{a}/a^0$, the corresponding hyperbolic cone $\mathsf {C}( \mathsf {K}_{\varepsilon })\subset \mathsf {H}_{\bar{\tau }}$ is given by

$$\begin{aligned} \mathsf {C}( \mathsf {K}_{\varepsilon })=\left\{ \, \bar{\tau }\frac{(1,\varvec{u})}{\sqrt{1-\varvec{u}^2}} \in \mathsf {H}_{\bar{\tau }} \,\bigg |\, \varvec{u}=\varvec{u}_{\varepsilon }+s\,(\varvec{\ell } -\varvec{u}_{\varepsilon } ),\ 0\le s<1, \ \varvec{\ell } \in \Theta _{\varepsilon } \right\} .\quad \end{aligned}$$

(C.4)

We note that as $\varepsilon \rightarrow 0$, the apex of $\mathsf {C}( \mathsf {K}_{\varepsilon })$ tends to lightlike infinity in the direction of $\varvec{\ell } _0$ and the opening angle tends to zero. In fact, for all $0\le s<1$ and $\varvec{\ell } \in \Theta _{\varepsilon },$ we have

$$\begin{aligned} \varvec{u}_{\varepsilon }(s, \varvec{\ell } ):=\varvec{u}_{\varepsilon }+s\,(\varvec{\ell } -\varvec{u}_{\varepsilon } )=\varvec{\ell } _0(1-\varepsilon (1-s))+s(\varvec{\ell } -\varvec{\ell } _0). \end{aligned}$$

(C.5)

Noting that $(\varvec{\ell } -\varvec{\ell } _0)^2=2(1-\varvec{\ell } \varvec{\ell } _0)\le 2\varepsilon$ and setting $\varvec{h}_{\varepsilon }(s, \varvec{\ell } ):=-\varepsilon ^{\frac{1}{2}}\varvec{\ell } _0(1-s)+s\varepsilon ^{-\frac{1}{2}}(\varvec{\ell } -\varvec{\ell } _0)$, we have

$$\begin{aligned}&\varvec{u}_{\varepsilon }(s, \varvec{\ell } )=\varvec{\ell } _0+\varepsilon ^{\frac{1}{2}} \varvec{h}_{\varepsilon }(s, \varvec{\ell } ), \end{aligned}$$

(C.6)

$$\begin{aligned}&|\varvec{h}_{\varepsilon }(s, \varvec{\ell } )|\le 3. \end{aligned}$$

(C.7)

Now, a simple computation using (C.5) gives

$$\begin{aligned} 1-\varvec{u}_{\varepsilon }(s, \varvec{\ell } )^2 =\varepsilon (1-s) \big \{ 2-\varepsilon (1-s) +2s(1-\varepsilon )(1-\varvec{\ell } \varvec{\ell } _0)\varepsilon ^{-1}\big \}. \end{aligned}$$

(C.8)

It is easy to see that $1\le \{\ldots \}\le 4$ and, therefore, we can find a function $(s,\varvec{\ell } )\mapsto g_{\varepsilon }(s, \varvec{\ell } )$ such that $\frac{\bar{\tau }}{2}\le g_{\varepsilon }(s, \varvec{\ell } )\le \bar{\tau }$ and

$$\begin{aligned} \bar{\tau }\frac{1}{\sqrt{1-\varvec{u}_{\varepsilon }(s, \varvec{\ell } )^2}}= \frac{g_{\varepsilon }(s, \varvec{\ell } )}{\sqrt{\varepsilon (1-s)}}. \end{aligned}$$

(C.9)

Thus, skipping the arguments of $g, \varvec{h}$ and setting $M:=\varepsilon ^{-\frac{1}{2}}$, $S:=g(1-s)^{-\frac{1}{2}}$, we have

$$\begin{aligned} \bar{\tau }\frac{(1,\varvec{u}_{\varepsilon }(s, \varvec{\ell } ))}{\sqrt{1-\varvec{u}_{\varepsilon }(s, \varvec{\ell } )^2}} =MS\!\cdot \!(1,\varvec{\ell } _0)+ S\!\cdot \!(0, \varvec{h}), \end{aligned}$$

(C.10)

where M takes values in $[\varepsilon _0^{-1/2}, \infty )$ and S in $[\frac{\bar{\tau }}{2}, \infty )$. Thus, we found a convenient parametrization of $\mathsf {C}( \mathsf {K}_{\varepsilon })$. We will use it to establish the relation (C.14) below, which ensures spacelike separation of $\mathsf {C}( \mathsf {K}_{\varepsilon })$ and $\mathcal {U}$ for sufficiently small $\varepsilon$.

As a preparation, let us show that there is a $c>0$ such that for sufficiently large M

$$\begin{aligned} (MS\!\cdot \!(1,\varvec{\ell } _0)+S\!\cdot \!(0, \varvec{h})-x)^2<-c, \end{aligned}$$

(C.11)

for all $x\in \mathcal {O}$, $S\in [\frac{\bar{\tau }}{2}, \infty )$ and $\varvec{h}$ within the above restrictions. Since $\overline{\mathcal {O}}\subset V$, there are constants $c_{\mathcal {O}}, c'_{\mathcal {O}}$ such that

$$\begin{aligned} 0<c_{\mathcal {O}} \le (x^0\pm |\varvec{x}|)\le c'_{\mathcal {O}}, \end{aligned}$$

(C.12)

uniformly in $x\in \mathcal {O}$. Moreover, due to (C.10), we have $(MS\!\cdot \!(1,\varvec{\ell } _0)+S\!\cdot \!(0, \varvec{h}))^2=\bar{\tau }^2$. Hence,

$$\begin{aligned} (MS\!\cdot \!(1,\varvec{\ell } _0)+S\!\cdot \!(0, \varvec{h})-x)^2&=\bar{\tau }^2-2MS(x^0-\varvec{x}\varvec{\ell } _0)-2S\!\cdot \!(0, \varvec{h})x+x^2\nonumber \\&\le -2MSc_{\mathcal {O}}+ 6Sc_{\mathcal {O}}'+ (c'_{\mathcal {O}})^2+\bar{\tau }^2, \end{aligned}$$

(C.13)

which proves (C.11).

Finally, let us show that there is a $c'>0$ such that for sufficiently large M, 

$$\begin{aligned} (MS\!\cdot \!(1,\varvec{\ell } _0)+S\!\cdot \!(0, \varvec{h})-x-\tau (1, \varvec{\ell } '))^2<-c', \end{aligned}$$

(C.14)

for all $\tau \in \mathbb {R}_+$, $\varvec{\ell } '\in \Theta$, $x\in \mathcal {O}$, $S\in [\frac{\bar{\tau }}{2}, \infty )$ and $\varvec{h}$ within the above restrictions. In view of (C.11), it suffices to note the estimate

$$\begin{aligned} \big (MS\!\cdot \!(1,\varvec{\ell } _0)+S\!\cdot \!(0, \varvec{h})-x\big )(1, \varvec{\ell } ')&= S( M(1-\varvec{\ell } _0\varvec{\ell } ')-\varvec{h}\varvec{\ell } ')-x(1,\varvec{\ell } ')\nonumber \\&\ge (\bar{\tau }/2)\big (M\varepsilon _0-3\big )-c_{\mathcal {O}}'. \end{aligned}$$

(C.15)

Thus, we have proven that $\mathcal {U}\subset \mathsf {C}( \mathsf {K}_{\varepsilon })^{\mathrm {c}}=\mathcal {C}(\mathsf {K}_{\varepsilon })^{\mathrm {c}}$ for $\varepsilon$ sufficiently small, depending on $\mathcal {O}$ and $\Theta$. $\square$