1 Introduction

The massless Nelson model is a time-honoured theoretical laboratory for the infrared aspects of QED. One of its variants, which we consider in this work, describes one non-relativistic massive particle (‘the electron’), interacting with massless scalar bosons (‘the photons’). The coupling between the electrons and photons is chosen in such a way that the model exhibits the infraparticle problem, i.e. it does not contain physical states describing the electron in empty space. In other words, the electron is always encircled by an ever larger halo of ever softer photons, and it is a challenge to mathematically describe the resulting composite object, usually called an infraparticle. Two milestones in rigorous understanding of this problem are works of Fröhlich [19, 21] and Pizzo [32, 33]. The latter two papers actually give a complete discussion of the infraparticle in the Nelson model and of its collisions with (hard) photons. Also collisions of an infraparticle with a Wigner-type particle (‘an atom’) in a Nelson model with two massive particles are under control [17]. However, scattering of several infraparticles appears steeply difficult in the conventional approach from [33], as discussed in detail in [17, Introduction]. One reason is that the approximating sequence of the infraparticle state from [33] and the proof of its convergence are technically quite intricate, which may be due to limited spectral information on the model available back then. Given intervening advances in spectral theory [1, 15, 16], we revisit the subject and propose a simpler approximating sequence of the infraparticle in the Nelson model. Its convergence to a non-trivial limit is relatively straightforward, given the currently available spectral ingredients. Needless to say, our discussion above only touched upon the broad topic of spectral and scattering theory in non-relativistic QED, see e.g. [3, 6, 8, 9, 13, 18, 36], and no systematic review is attempted here.

To explain our construction, let us recall the definition of the Nelson model. The Hilbert space of the model is \({\mathcal {H}}=L^2({\mathbb {R}}^3_x; {\mathcal {F}})\), where \({\mathcal {F}}\) is the symmetric Fock space over \(L^2({\mathbb {R}}^3_k)\). For introductory material on the theory of Fock spaces, we refer the reader to [34, Sect. X.7]. Thus, we will treat \(\psi \in {\mathcal {H}}\) as \({\mathcal {F}}\)-valued square-integrable functions \(\{\psi (x)\}_{x\in {\mathbb {R}}^3}\), whose scalar product has the form

$$\begin{aligned} \langle \psi _1,\psi _2\rangle _{{\mathcal {H}}}=\int d^3x\, \langle \psi _{1}(x), \psi _{2}(x)\rangle _{{\mathcal {F}}}. \end{aligned}$$
(1.1)

The creation and annihilation operators on \({\mathcal {F}}\) are denoted by \(a^{(*)}(f)\), \(f\in L^2({\mathbb {R}}_k^3)\), and their sharp variants by \(k\mapsto a^{(*)}(k)\). We will also occasionally write

$$\begin{aligned} \Phi (f):=a^*(-if)+a(-if). \end{aligned}$$
(1.2)

The Hamiltonian of the Nelson model has the form

$$\begin{aligned} H=\frac{(-i\nabla _{x})^2}{2}+H_{\textrm{f}}+a(v_x)+a^*(v_x). \end{aligned}$$
(1.3)

Here, x and \(-i\nabla _x\) are the position and momentum operators on \(L^2({\mathbb {R}}^3_x)\), \((H_{\textrm{f}}, P_{\textrm{f}}):=(\textrm{d}\Gamma (|k|), \textrm{d}\Gamma (k))\) denote the energy-momentum operators of non-interacting photons and \(v_x(k)=v(k) e^{-ik\cdot x}\), where \(v(k):=\lambda \frac{\chi _{\kappa }(k)}{\sqrt{2|k|}}\) and \(|\lambda |\in (0,\lambda _0]\) is the coupling constant, whose maximal value \(\lambda _0\) will be sufficiently small but nonzero. Here, \(\chi _{\kappa }\in C^{\infty }_0({\mathbb {R}}^3)\) is a smooth approximate characteristic function of the ball of radius \(\kappa =1\).Footnote 1 We choose this function rotation invariant, supported in the ball of radius \(\kappa \) and equal to one on a ball of a slightly smaller radius \((1-\varepsilon _0)\kappa \) for some \(0<\varepsilon _0<1\). By the Kato–Rellich theorem, H is a self-adjoint operator on \(D(\frac{1}{2}(-i\nabla _{x})^2+H_{\textrm{f}} )\). This elementary observation dates back to [30], for a textbook discussion in a similar model we refer to [36, Sect. 13.3]. The origin of the infrared problem lies in the fact that \(v(k)/|k|\notin L^2({\mathbb {R}}^3_k)\), as will be recalled later in Sect. 2.

Recalling that the model is translation invariant, we denote by \(\{H_p\}_{p\in {\mathbb {R}}^3}\) the usual fibre Hamiltonians acting on the fibre Fock space \({\mathcal {F}}_{\textrm{fi}}\), satisfying

$$\begin{aligned} H=\Pi ^*\left( \int ^{\oplus }d^3p\, H_p\, \right) \Pi , \quad \Pi =Fe^{iP_{\textrm{f}} \cdot x}, \end{aligned}$$
(1.4)

where F is the Fourier transform in the x variable. In our construction of infraparticle scattering states we will identify the fibre Fock space \({\mathcal {F}}_{\textrm{fi}}\) with the physical Fock space \({\mathcal {F}}\) which is the reason for the appearance of the unitary \(\Pi \) explicitly in formula (1.8). After this identification, the fibre Hamiltonians are the following self-adjoint operators on \(D(P_{\textrm{f}}^2+H_{\textrm{f}})\subset {\mathcal {F}}\)

$$\begin{aligned} H_p:=\frac{1}{2}(p-P_{\textrm{f}})^2+H_{\textrm{f}}+a^*(v)+a(v), \quad p\in {\mathbb {R}}^3. \end{aligned}$$
(1.5)

We denote the infimum of the spectrum of \(H_p\) by \(E_p\). One manifestation of the infraparticle problem is that \(E_p\) is not an eigenvalue. This has been established in considerable generality in [11, 21, 32]. For \( p\in S\), where

$$\begin{aligned} S:=\{\, p'\in {\mathbb {R}}^3 \,|\, |p'| {<} 1/3\,\} \end{aligned}$$
(1.6)

and \(\lambda _0\) sufficiently small we know in addition from [1] that \(p\mapsto E_p\) is real analytic and \(|\nabla E_p|<1/2\) for \(p\in S\) (cf. Lemma 3.1). It is also well-known that the modified Hamiltonian \(H_p^{\textrm{w}}\), obtained from \(H_p\) by the Bogolubov transformation

$$\begin{aligned} a^{(*)}(k)\mapsto a^{(*)}(k)-f_p(k), \quad f_p(k):=\lambda \frac{\chi _{\kappa }(k)}{\sqrt{2|k|}}\frac{1}{|k|(1-e_k\cdot \nabla E_{p} )},e_k:=k/|k|, \nonumber \\ \end{aligned}$$
(1.7)

is self-adjoint on \(D(P_{\textrm{f}}^2+H_{\textrm{f}})\) and \(E_p\) is its eigenvalue at the bottom of the spectrum corresponding to an eigenvector \(\phi _p\) [32]. (Its phase is chosen in the following in accordance with [15, Definition 5.2].) Such a change in the character of \(E_p\) is possible because \(f_p\notin L^2({\mathbb {R}}^3)\), and hence, the Bogolubov transformation (1.7) is not unitarily implementable.

After these preparations we are ready to define the approximating sequences of the infraparticle states. Motivation for this formula is given in Sect. 2, and in Conclusions, we relate it to the Faddeev–Kulish approach. For any \(h\in C^{\infty }_0({\mathbb {R}}^3)\) supported in S and any time parameter \({t\in {\mathbb {R}}}\), we set

$$\begin{aligned}&\psi _t(x):=e^{iHt}e^{-i P_{\textrm{f}}\cdot x} \nonumber \\&\quad \times {\frac{1}{(2\pi )^{3/2}}} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p (e^{-i|k| t+ik\cdot x}-1) \big ) \phi _p, \end{aligned}$$
(1.8)
$$\begin{aligned}&\gamma (p,x,t):= \int d^3k f_p(k)^2 \sin (|k|t- k\cdot x ), \end{aligned}$$
(1.9)

where \(W(g):=e^{a^*(g)-a(g)}, g\in L^2({\mathbb {R}}^3_k)\), is a Weyl operator on \({\mathcal {F}}\). It is well-defined for \(g(k):=f_p{(k)} (e^{-i|k| t+ik\cdot x}-1)\) for any \((t,x)\in {\mathbb {R}}^4\). This is due to the fact that \(|e^{-i|k| t+ik\cdot x}-1|\le |k|(|x|+|t|)\) and hence \(k\mapsto f_p(k)(e^{-i|k| t+ik\cdot x}-1)\) is square integrable, unlike \(f_p\), cf. Lemma E.1. The integral in (1.8) is well-defined as a Bochner integral in \({\mathcal {F}}\), since \(S\ni p\mapsto \phi _p\) is Hölder continuous in norm by [32] (which can also be seen by [15, formulas (1.8), (A.4) and Corollary 5.6] combined with Lemma C.3 below). This integral belongs to \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) by Lemma 4.5. Our main result is the following:

Theorem 1.1

There is such \(\lambda _0>0\) that the following holds: For any \(t\in {\mathbb {R}}\), the vector \(\psi _t\) given by (1.8) belongs to \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\). The derivative \(\partial _t\psi _t\) exists in norm in \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\), and we have

$$\begin{aligned}{} & {} \partial _t\psi _t { (x)}=e^{iHt} e^{-iP_{\textrm{f}}\cdot x } \nonumber \\{} & {} \quad \times \frac{1}{(2\pi )^{3/2}} \int d^3p \, e^{i (p \cdot x -E_pt )} e^{i \gamma (p,x,t)} i \gamma _{\textrm{int}}(p,x,t) h(p) W(f_p (e^{-i|k| t+ik\cdot x }-1)) \phi _p, \nonumber \\ \end{aligned}$$
(1.10)

where \(\gamma _{\textrm{int}}(p,x,t):= 2 \int d^3k\, f_p(k)^2(|k|-k\cdot \nabla E_p) \cos (|k|t-k\cdot x)\) is rapidly decreasing in the region \(|x|/t<1\) (cf. Lemma 4.7). Furthermore,

$$\begin{aligned} \int _0^{\infty } \hbox {d}t\, \Vert \partial _t\psi _t\Vert _{{\mathcal {H}}}<\infty , \end{aligned}$$
(1.11)

hence \(\psi ^{+}:=\lim _{t\rightarrow \infty } \psi _t\) exists in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\). For \(h\ne 0\) and \({|\lambda |\in (0, \lambda _0]}\) sufficiently small, \(\psi ^{+}\ne 0\). Analogous statements hold for incoming scattering states.

The most remarkable part of the theorem is the explicit formula for \(\partial _t\psi _t\) given in (1.10). It can be anticipated by formal computations on \({\mathcal {F}}\) noting the key relation

$$\begin{aligned} \begin{aligned}{} & {} T(p,x,t)^{*} \left( -i\nabla _x-P_\textrm{f}\right) T(p,x,t)= -i\nabla _x - P_\textrm{f}^\textrm{w}, \\{} & {} \quad T(p,x,t):= W(f_p (e^{-i|k| t+ik\cdot x }-1)) e^{i \gamma (p,x,t)}, \end{aligned} \end{aligned}$$
(1.12)

where \(P_\textrm{f}^\textrm{w}\) is obtained from \(P_{\textrm{f}}\) via the Bogolubov transformation (1.7). Relation (1.12) allows to reconstruct \(H^{\textrm{w}}_p\) in front of \(\phi _p\) and make use of \(H^{\textrm{w}}_p\phi _p=E_p\phi _p\). It dictates our choice of the phase \(\gamma \), and it is noteworthy that the resulting \(\gamma _{\textrm{int}}\) enjoys a rapid decay in t in the physical region of velocities of the electron. This coincidence suggests that our approximating vector (1.8) captures optimally the asymptotic dynamics of the Nelson model in the infrared regime. The decay of \(\gamma _{\textrm{int}}\) is the driving force of our convergence argument based on the Cook method [10, 35]. It also allows for a simple proof of non-triviality of the limit for small \(|\lambda |\).

Given formula (1.10) and the above remarks, it may seem very easy to prove the theorem. But it should be kept in mind that estimate (1.11) must hold in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\), which involves the integral over the whole space, cf. formula (1.1). To control this integral, we use the stationary phase method, which generates derivatives w.r.t. p up to the second order (cf. Lemma 4.1). Since differentiability of \(p\mapsto \phi _p\) is not settled, we have to approximate \(\phi _p\) with \(\phi _{p,\sigma }\), which come from the Nelson model with an infrared cut-off \(\sigma >0\) in the interaction. The function \(p\mapsto \phi _{p,\sigma }\) is differentiable, and its derivatives up to the second order have only a mild infrared divergence of the form

$$\begin{aligned} \Vert \partial _{p}^{\alpha } \phi _{p,\sigma }\Vert _{{\mathcal {F}}}\le c\sigma ^{-\delta _{\lambda _0}}, \quad {|\alpha |=0,1,2,} \end{aligned}$$
(1.13)

where \(\delta _{\lambda _0}>0\) tends to zero with \(\lambda _0\rightarrow 0\). This estimate, and similar bounds for the wave functions of \(\phi _{p,\sigma }\), rely on technical advances from [15, 16]. Thus, at our present level of understanding, we can eliminate the infrared cut-off from the formulation of Theorem 1.1, but not from its proof. As mentioned above, to eliminate the cut-off also from the proof it seems necessary to establish differentiability of \(p\mapsto \phi _p\). For some ideas in this direction, we refer to [15, formula (1.9)] and a cancellation of infrared singularities conjectured in this formula.

This paper is organized as follows: In Sect. 2, we provide motivation for our infraparticle ansatz (1.8). In Sect. 3, we give some technical information, in particular about the model with infrared cut-off. Section 4 is devoted to the proof of Theorem 1.1. In Conclusions, we provide a brief comparison of our infraparticle states with the Faddeev–Kulish approach. More technical parts of the discussion are postponed to Appendices.

2 Motivation for the Infraparticle Ansatz (1.8)

The fibre decomposition and the associated transformation, both seen in (1.4), are due to Lee, Low and Pines [27] and have been used ever since. We find it convenient to phrase that transformation as a superposition instead. To convey the quite trivial idea, let us first consider the simpler case of functions \(\Psi \in L^2({\mathbb {R}}_x)\), which can be written as \(\Psi =(2\pi )^{-1/2}\int dp\, \Psi _p\) where \(\Psi _p=\Psi _p(x)\) is \(\Psi _p(x)=\hat{\Psi }_p e^{ipx}\). While \(\Psi \mapsto \hat{\Psi }\) is the Fourier transform, the integral itself is a superposition of improper elements of \(L^2({\mathbb {R}}_x)\). The former transformation is more precise, the latter is closer to physical intuition because it displays \(\Psi \) as the superposition of plane waves.

In the case of \({\mathcal {H}}=L^2({\mathbb {R}}_x;{\mathcal {F}})\), the decomposition of \(\Psi \in {\mathcal {H}}\) is

$$\begin{aligned} \Psi ={ (2\pi )^{-3/2} } \int _{{\mathbb {R}}^3} d^3p \, \Psi _{p}, \end{aligned}$$
(2.1)

where \(\Psi _p\) is an improper element of \({\mathcal {H}}\). It is singled out by its character \(y\mapsto e^{-ip\cdot y}\) (in the sense of representation theory) of the (abelian) translation group \(e^{-iP\cdot y}\), \(P=-i\nabla _x+P_{\textrm{f}}\),

$$\begin{aligned} e^{-iP\cdot y} \Psi _{p}= e^{-ip\cdot y} \Psi _{p}. \end{aligned}$$
(2.2)

An informal expression for \(\Psi _{p}\) is provided by the Wigner projection onto the isotypical component associated with the character:

$$\begin{aligned} \Psi _{p}= & {} { (2\pi )^{-3/2} } \int d^3y\, \overline{e^{-ip\cdot y} } e^{-iP\cdot y} \Psi ={ (2\pi )^{-3/2} } \int d^3y\, e^{-i(P-p)\cdot y} \Psi . \end{aligned}$$
(2.3)

Indeed, by \(\int d^3p\, e^{ip\cdot y} =(2\pi )^3 \delta (y)\), we have

$$\begin{aligned} { (2\pi )^{-3/2}} \int d^3p\, \Psi _{p} =\int d^3y\, \delta (y)e^{-iP\cdot y} \Psi =\Psi . \end{aligned}$$
(2.4)

Let us also note that \(\Psi _{p}\) ”\(\in \)” \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) takes values \(\Psi _{p}(x)\) in \({\mathcal {F}}\) for \(x\in {\mathbb {R}}^3\). They are related to one another by

$$\begin{aligned} \Psi _{p}(x-y)=(e^{-i(-i\nabla _x) \cdot y} \Psi _{p})(x)=(e^{-i(p-P_{\textrm{f}}) \cdot y} \Psi _{p})(x)= e^{-i(p-P_{\textrm{f}}) \cdot y} \Psi _{p}(x)\nonumber \\ \end{aligned}$$
(2.5)

because \(P_{\textrm{f} }\) acts on \({\mathcal {F}}\) alone. In particular, setting x to zero and then renaming y to \(-x\), we get

$$\begin{aligned} \Psi _{p}(x)=e^{i(p-P_{\textrm{f}} )\cdot x } \Psi _{p}(0). \end{aligned}$$
(2.6)

For a rigorous treatment, it is better to restate (2.1) as a unitary map

$$\begin{aligned} \Pi : {\mathcal {H}}\rightarrow L^2({\mathbb {R}}_p; {\mathcal {F}}), \quad \Psi \mapsto (\Pi _{p} \Psi )_{ p \in {\mathbb {R}}^3}. \end{aligned}$$
(2.7)

This is achieved by evaluation at, say, \(x=0\)

$$\begin{aligned} \Pi _{p}\Psi =\Psi _{p}(0). \end{aligned}$$
(2.8)

By (2.1), (2.6) the inverse map of \(\Pi \), i.e. \(\Pi ^{-1}:( { \Psi _{p}(0) } )_{p\in {\mathbb {R}}^3} \mapsto \Psi \) is

$$\begin{aligned} \Psi = { (2\pi )^{-3/2}} \int d^3p\, e^{i(p-P_{\textrm{f}})\cdot x } { \Psi _{p}(0) }. \end{aligned}$$
(2.9)

We note that (2.3) can also be written as

$$\begin{aligned} \Psi _p(x)=(2\pi )^{-3/2}\int d^3y\, e^{ip\cdot y} e^{-iP_{\textrm{f}}\cdot y }\Psi (x-y) \end{aligned}$$
(2.10)

because of \((e^{-iP\cdot y} \Psi )(x)=e^{-iP_{\textrm{f}}\cdot y } \Psi (x-y)\). Thus

$$\begin{aligned} \Pi _p\Psi =\Psi _p(0)=(2\pi )^{-3/2} \int d^3x\, e^{-ip\cdot x} e^{iP_{\textrm{f}}\cdot x} \Psi (x) \end{aligned}$$
(2.11)

by the substitution \(y=:-x\), which is possible because x has been disposed of by setting \(x=0\) first. We conclude that \(\Pi _p\Psi =F(e^{iP_{\textrm{f}}\cdot x }\Psi )\).

The understanding of the ansatz (1.8) benefits from a comparison with the van Hove model. Compared to the Nelson model, the (dynamical) electron is replaced by an external source. More formally, the Hilbert space is \({\mathcal {F}}\) and the Hamiltonian is

$$\begin{aligned} H=\int d^3k \, \omega (k) (a^*(k) - \bar{f}(k)) (a(k) - f(k)). \end{aligned}$$
(2.12)

We observe that it resembles (1.3) once x and \(\nabla _x\) are omitted and \(\omega (k)=|k|\) (though up to zero point energy irrelevantly differing by \(\int d^3k\,\omega (k) |f(k)|^2\)). Regarding \(f\), let us first assume that \(f\in L^2({\mathbb {R}}^3_k)\) and comment on the physical choice \(f(k)=-v(k)/|k|\notin L^2{ ({\mathbb {R}}^3_k)}\) later on.

The free Hamiltonian \(H_0\), corresponding to \(f=0\), and the Weyl operators \(W(g)\) (\(g\in L^2({\mathbb {R}}^3_k)\)) satisfy:

$$\begin{aligned} W(g_1)W(g_2)= & {} e^{-i \textrm{Im}{ \langle } g_1,g_2{ \rangle } } W(g_1+g_2), \end{aligned}$$
(2.13)
$$\begin{aligned} e^{-itH_0} W({ g})= & {} W(e^{-it\omega } { g}) e^{-itH_0}; \end{aligned}$$
(2.14)

moreover, \(H_0\) is unitarily equivalent to the Hamiltonian

$$\begin{aligned} H=W(f) H_0 W(-f), \end{aligned}$$
(2.15)

since \(f\in L^2({\mathbb {R}}^3_k)\) and thus \(W(f)\) well-defined. In particular, \(W(f)\Omega \) is the ground state of H:

$$\begin{aligned} HW(f)\Omega =E_0W(f)\Omega \end{aligned}$$
(2.16)

with \(E_0=0\) for the above choice of zero point energy. Equations (2.13, 2.14) imply the identities:

$$\begin{aligned} e^{-itH}= & {} e^{-i\langle f, \sin (\omega t) f\rangle } W((1-e^{-it\omega }) f) e^{-itH_0}, \end{aligned}$$
(2.17)
$$\begin{aligned} e^{-itH} W(g)= & {} e^{2i \textrm{Im}\langle f, (1-e^{-it\omega } ) g\rangle } W( e^{-it\omega }g) e^{-itH}. \end{aligned}$$
(2.18)

Using (2.17), we get

$$\begin{aligned} e^{-itH} \Omega= & {} e^{-itE_0} e^{-i\langle f, \sin (\omega \, t) f\rangle } W((1-e^{-it\omega }) f) \Omega , \end{aligned}$$
(2.19)

which describes the evolution of the bare vacuum \(\Omega \). Somewhat more suggestively, it is restated as

$$\begin{aligned} e^{-itH } \Omega = e^{-itE_0} e^{-{ 2}i\langle f, \sin (\omega \, t) f\rangle } W( -e^{-it\omega } f) W(f) \Omega . \end{aligned}$$
(2.20)

In this guise, it describes the approach of the unperturbed vacuum \(\Omega \) to the perturbed ground state \(W(f) \Omega \) as \(t\rightarrow { +}\infty \) (up to a numerical factor \(e^{-\frac{1}{2}\Vert f\Vert _2^2} \)). In fact, we note that \(e^{-it\omega } f\rightarrow 0\) weakly in \(L^2({\mathbb {R}}^3_k)\) as \(t\rightarrow \infty \), whence \(W( e^{-it\omega } f) \rightarrow e^{-\frac{1}{2}\Vert f\Vert _2^2}\) in the weak topologyFootnote 2 of operators on \({\mathcal {F}}\). The decreased norm can be attributed to photons lost at infinity in \({\mathbb {R}}^3_x\).

Equation (2.18), when applied to \(\Psi \in {\mathcal {F}}\), states that out of the trajectory \(e^{-itH} \Psi \) another one can be obtained by adding a coherent bunch of freely moving bosons by means of \(W(e^{-it\omega } g)\); in fact up to phase, the trajectory \(e^{-itH}W(g)\Psi \) results. For example, we can choose \(\Psi =W(f)\Omega \), i.e. the perturbed ground state, in which case

$$\begin{aligned} { e^{-itH} W(g)W(f)\Omega =e^{2i \textrm{Im}\langle f, (1-e^{-it\omega })g\rangle } W(e^{-it\omega } g) e^{-itE_0} W(f)\Omega . } \end{aligned}$$
(2.21)

For \(g=-f\), we simply recover (2.20). The unperturbed ground state \(\Omega \) is referred to as bare and the perturbed one \(W(f)\Omega \) then arises in the same picture by the dressing transformation W(f). In its own “infrared” picture the perturbed ground state is still given by \(\Omega \), because the Weyl operator intertwines exactly between H and \(H_0\).

The origin of the infrared problem, which arises when \(f\notin L^2({\mathbb {R}}^3_k)\), is now manifest: So to speak, the ground state \(W(f)\Omega \) in (2.16) leaves the Fock space \({\mathcal {F}}\). While \(\Psi =W(f)\Omega \) is not well-defined, a trajectory in \({\mathcal {F}}\) can be defined via (2.21), provided that \(g+f\in L^2({\mathbb {R}}^3_k)\). To this end, the pair of Weyl operators seen there or in (2.20) should be merged to one, as done in (2.19). Then, by restating (2.21) for \(g+f=0\) as follows

$$\begin{aligned} \Omega =e^{itH} e^{-itE_0} e^{i\textrm{Im}\langle f, e^{-it\omega } f\rangle } W((1-e^{-it\omega } )f)\Omega \end{aligned}$$
(2.22)

we note a similar structure as in our infraparticle vector (1.8).

These conclusions can be transposed to the Nelson model, which fibre-wise resembles the van Hove model of coupling \(f(k)=-v(k)/|k|\notin L^2{ ({\mathbb {R}}^3_k)}\). In the Nelson model, the ground state of \(H^{\textrm{w}}_{p}\) of total momentum p is \(\phi _{p} \) in its own (infrared) picture,

$$\begin{aligned} H^{W}_p\phi _p=E_p\phi _p. \end{aligned}$$
(2.23)

That state is perturbatively close to, but no longer identical to \(\Omega \), because no Weyl operator removes the interaction terms in (1.3) exactly.

In the bare picture that state is \(W({ -}f_p) \phi _p\), where

$$\begin{aligned} f_p(k)={ -}\frac{f(k)}{1-e_k\cdot \nabla E_p}. \end{aligned}$$
(2.24)

Since \(f_p\notin L^2({\mathbb {R}}^3_k)\), this vector is not well-defined (in contrast to \(\phi _p\)).

Collecting fibres, cf. (2.9), we obtain (still not well-defined) states on the mass shell \(p\mapsto E_p\),

$$\begin{aligned} \phi (x)={ (2\pi )^{-3/2} } \int d^3p\, h(p) e^{i(p-P_{\textrm{f}} )\cdot x } W({ -}f_p)\phi _p, \end{aligned}$$
(2.25)

where the support of h is contained in the set S, cf. (1.6). Its trajectory \(\phi _t=e^{-itH} \phi \) is

$$\begin{aligned} \phi _t(x)={ (2\pi )^{-3/2} }\int d^3p \, h(p) e^{i(p\cdot x - E_p t )} e^{-iP_{\textrm{f}}\cdot x } W({ -}f_p)\phi _{p}. \end{aligned}$$
(2.26)

The goal is to add a bunch of photons to \(\phi _t\) in a way that is simple and explicit, though not strictly compatibly with \(e^{-iHt}\) as in (2.18), and yet in such a way that:

  • unlike (2.26), the resulting state \(\Psi _t=\Psi _t(x)\) lies in \({\mathcal {H}}\),

  • the addition is asymptotically compatible with dynamics, in the sense that the limit

    $$\begin{aligned} \lim _{t\rightarrow \infty } e^{itH} \Psi _t={ \psi ^+} \end{aligned}$$
    (2.27)

    exists. That in fact means that \(e^{-iHt}{ \psi ^+}\) has \(\Psi _t\) as its explicit asymptote.

In line with (2.18), its interpretation and its use for \(g=-f\), we modify (2.26) to

$$\begin{aligned} \Psi _t(x)={ (2\pi )^{-3/2} }\int d^3p\, h(p) e^{2i\tilde{\gamma }} W(e^{-i\omega t} f_p) e^{i(p\cdot x - E_p t )} e^{-iP_{\textrm{f}}\cdot x } W({ -}f_p)\phi _{p}, \nonumber \\ \end{aligned}$$
(2.28)

where the phase \(\tilde{\gamma }=\tilde{\gamma }(x,p,t)\) is going to be chosen in a moment. By

$$\begin{aligned} e^{-i P_{\textrm{f}}\cdot x } W(g)=W(e^{-ik\cdot x} g) e^{-iP_{\textrm{f}}\cdot x }, \end{aligned}$$
(2.29)

cf. (2.13), (2.14), we get

$$\begin{aligned} \Psi _t(x)={ (2\pi )^{-3/2} }\int d^3p\, h(p) e^{2i\tilde{\gamma }} e^{i(p\cdot x-E_p t)} e^{-iP_{\textrm{f}} \cdot x } W(e^{i(k\cdot x - \omega t)} f_p ) W({ -}f_p)\phi _p. \nonumber \\ \end{aligned}$$
(2.30)

Now, we choose the phase by comparing (2.30) with (2.21). We recall (2.21)

$$\begin{aligned} e^{-itH} W(g)W(f)\Omega= & {} e^{-itE_0} e^{2i \textrm{Im}\langle f, (1-e^{-it\omega } ) g\rangle } W( e^{-it\omega }g) W(f)\Omega , \end{aligned}$$
(2.31)

and make substitutions \(g\rightarrow e^{ik\cdot x}f_{p}\), \(f\rightarrow { -}f_p\). This suggests

$$\begin{aligned}{} & {} \tilde{\gamma }=\textrm{Im}\langle f_p, -(1-e^{-i(\omega t-k\cdot x) }) f_p \rangle =\textrm{Im}\langle f_p, e^{-i(\omega t -k\cdot x ) } f_p\rangle \nonumber \\{} & {} \quad ={-\langle f_p,\sin (\omega t-k\cdot x) f_p\rangle }. \end{aligned}$$
(2.32)

Finally, we merge the Weyl operators in (2.30),

$$\begin{aligned} W(e^{ -i(\omega t-k\cdot x ) } f_p)W({ -} f_p)=e^{-i\tilde{\gamma }}W({ -} (1-e^{-i(\omega t -k\cdot x ) } )f_p ). \end{aligned}$$
(2.33)

We note that the argument of the last Weyl operator lies in \(L^2({\mathbb {R}}^3_k)\) for each x, and we obtain from (2.30)

$$\begin{aligned} \Psi _t(x)={ (2\pi )^{-3/2} }\int d^3p\, h(p) e^{i\tilde{\gamma }} e^{i(p\cdot x-E_p t)} e^{-iP_{\textrm{f}} \cdot x } W({ -} (1-e^{-i(\omega t -k\cdot x ) } )f_p )\phi _p.\nonumber \\ \end{aligned}$$
(2.34)

The result is similar but not identical to (1.8): The phases (1.9) and (2.32) have opposite signs. The first phase is so chosen to make identity (1.12) possible. For comparison, had the same choice been made for the (exactly solvable) van Hove model, then the approximant to \(e^{-itH}\Omega \) would not be \(e^{-itH}\Omega \) itself, in the form of the r.h.s. of (2.19), but \(e^{-2i\tilde{\gamma }}e^{-itH}\Omega \), leading to \(\psi _t=e^{-2i\tilde{\gamma }}\Omega \) as a counterpart to (1.8). By \(\tilde{\gamma }\rightarrow 0\) (\(t\rightarrow \infty \)), we still have \(\psi _t\rightarrow \Omega \), but \(\partial _t\psi _t=-2i(\partial _t\tilde{\gamma })\psi _t=i\tilde{\gamma }_{\textrm{int}}\Psi _t\) with \(\tilde{\gamma }_{\textrm{int}}=2\langle f_p, \omega \textrm{cos}(\omega t) f_p\rangle \), in line with (1.10).

3 Preliminaries

Recall that \(\{H_p\}_{p\in {\mathbb {R}}^3}\) are the fibre Hamiltonians (1.5) and let \(\{H_{p,\sigma }\}_{p\in {\mathbb {R}}^3}\) be their counterparts at an infrared cut-off \(0<\sigma \le \kappa \). This means that the form factor \(v\), defined below (1.3), is replaced with \(v^{\sigma }\) given by

$$\begin{aligned} v^{\sigma }(k):=\lambda \frac{\chi _{[\sigma ,\kappa )}(k) }{\sqrt{2|k|} }. \end{aligned}$$
(3.1)

Here, \(\chi _{[\sigma ,\kappa )}(k)={\textbf{1}}_{{\mathcal {B}}'_{\sigma } }(k)\chi _{\kappa }(k)\), \({\mathcal {B}}'_{\sigma }\) is the complement of the ball of radius \(\sigma \) and \({\textbf{1}}_{\Delta }\) is the characteristic function of a set \(\Delta \). We remark that \(\{H_{p,\sigma }\}_{p\in {\mathbb {R}}^3}\) act on a dense domain in \({\mathcal {F}}\), that is, no infrared cut-off is introduced on the Fock space. We will work in the range of parameters for which the technical results of [15,16,17] hold. That is,

$$\begin{aligned} { |\lambda |\le \lambda _0}, \quad \sigma \in (0,\kappa _{\lambda _0}], \quad p\in S:=\{\, p'\in {\mathbb {R}}^3 \,|\, |p'|{<} 1/3\}, \end{aligned}$$
(3.2)

where \(\lambda _0\) is sufficiently small and \(0<\kappa _{\lambda _0}\le \kappa \). As the fibre Hamiltonians \(H_{p}, H_{p,\sigma }\) are bounded from below, we can define

$$\begin{aligned} E_{p}:=\textrm{inf}\,\varvec{\sigma }(H_p), \quad E_{p,\sigma }:= \textrm{inf}\,\varvec{\sigma }(H_{p,\sigma }), \end{aligned}$$
(3.3)

where \(\varvec{\sigma }\) denotes the spectrum. (Occasionally, we will write \(E_p^{(\lambda )}\), \(E_{p,\sigma }^{(\lambda )}\) etc. if the dependence on \(\lambda \) will play a role.) \(E_p\) enters our definition of the infraparticle state (1.10), and our analysis relies on the following result:

Lemma 3.1

[1]. The function \(S\times {\mathcal {B}}_{\lambda _0}\ni (p,\lambda ) \mapsto E_{p}^{(\lambda )}\) is real-analytic and non-constant. It satisfies \(|\nabla _p E^{{(\lambda )}}_{p}|\le 1/2\) and its Hessian matrix in the p-variable is bounded from below for \(p\in S\) by a positive constant, uniformly in \(\lambda \).

We recall that the modified Hamiltonians \(H^{\textrm{w}}_p\) are obtained from \(H_p\) by the Bogolubov transformation (1.7) and their ground states are denoted \(\phi _p\). Similarly, the modified Hamiltonians \(H^{\textrm{w}}_{p,\sigma }\) are obtained from \(H_{p,\sigma }\) by the transformation

$$\begin{aligned} a^{(*)}(k)\mapsto a^{(*)}(k)-f_{p,\sigma }(k), \quad f_{p,\sigma }(k):=\lambda \frac{\chi _{[\sigma ,\kappa )}(k)}{\sqrt{2|k|}}\frac{1}{|k|(1-e_k\cdot \nabla E_{p,\sigma } )} \nonumber \\ \end{aligned}$$
(3.4)

and their ground states are denoted \(\phi _{p,\sigma }\). Both \(\phi _p\) and \(\phi _{p,\sigma }\) are in the domain of any power of \(H_{\textrm{f}}\) (cf. Lemma C.3) and in addition \(\phi _{p,\sigma }\) are in the domain of any power of the number operator \(N:=\textrm{d}\Gamma (1)\), cf. Lemma C.2. For a choice of the phases of \(\phi _p, \phi _{p,\sigma }\) as in [15, Definition 5.2], the following estimate holds

$$\begin{aligned} \Vert (H_{\textrm{f}})^{\ell } (\phi _{p}-\phi _{p,\sigma })\Vert _{{\mathcal {F}}}\le c\sigma ^{1/5}, \quad p\in S, \quad \ell \in {\mathbb {N}}_0, \end{aligned}$$
(3.5)

provided that \(\lambda _0>0\) is readjusted for each \(\ell \). It is well-known for \(\ell =0\) [32, 17, Corollary 5.6 (a)] and for \(\ell \in {\mathbb {N}}\), it is shown in Appendix C. We will also need the following lemma:

Lemma 3.2

Fix \(\ell _1,\ell _2\in {\mathbb {N}}_0\). Then, there exists \(\tilde{\lambda }_0>0\) and a positive function \( { [-\tilde{\lambda }_0, \tilde{\lambda }_0]} \ni \lambda _0\mapsto \delta _{\lambda _0}\) s.t. \(\lim _{\lambda _0\rightarrow 0} \delta _{\lambda _0}=0\) with the following property: For any fixed \(\lambda _0\in { [-\tilde{\lambda }_0, \tilde{\lambda }_0]}\) and all \(\sigma \in (0, \kappa _{\lambda _0}]\),

$$\begin{aligned} \Vert H_{\textrm{f}}^{\ell _1}N^{\ell _2}\partial _p^{\alpha } \phi _{p,\sigma }\Vert _{{\mathcal {F}}}\le \frac{c}{\sigma ^{\delta _{\lambda _0} } } \quad \text { for }\quad |\alpha |=0,1,2. \end{aligned}$$
(3.6)

The constant c is independent of \(p, \sigma , \lambda \) within the restrictions (3.2) but may depend on \(\ell _1, \ell _2\).

In Appendix B, we show how to extract the proof of Lemma 3.2 from [15, 16]. We remark that Lemma 3.1, bound (3.5), and Lemma 3.2 are the technical basis for our discussion in the next section.

We remark that a possible dependence of the constants c in (3.6) and (3.5) on \(\ell ,\ell _1,\ell _2\) does not cause complications, because it suffices to consider \(\ell ,\ell _1,\ell _2\le L\) for some finite L fixed throughout the proof. This can be seen from the discussion below (4.55) and from the proof of Lemma 4.4.

Notation. As we will discuss only outgoing scattering states, we set \(t\ge 1\). We denote by c numerical constants which may change from line to line. These constants are independent of \(\sigma \), p, \(\lambda \), t, x within the assumed restrictions, but may depend on h, \(\lambda _0\), \(\varepsilon _0\), where \(\varepsilon _0\) was defined below  (1.3). The functions denoted \(\lambda _0\mapsto \delta _{\lambda _0}\) are positive and satisfy \(\lim _{\lambda _0\rightarrow 0} \delta _{\lambda _0}=0\). They are independent of \(\sigma \), p within the assumed restrictions but may depend on \(\varepsilon _0\). These functions may change from line to line.

4 Infraparticle States

The goal of this section is to provide a proof of Theorem 1.1. Our main tool will be the stationary phase method. The estimates suitable for our purposes are stated in the following lemma, which is proven in Appendix D.

Lemma 4.1

Let \(p\mapsto g(p)\in {\mathcal {F}}\) be weakly infinitely differentiable on some dense domain and compactly supported in S. Let \(c_0\) be s.t. \(|\nabla E_p|<c_0<1\) for \(p\in \textrm{supp}\,{g}\). Then, for any \(0\le \varepsilon \le 1/2\),

$$\begin{aligned}{} & {} \bigg ( \int _{|x|/t\le c_0} d^3x\,\bigg \Vert \int d^3p \, e^{i(p\cdot x-{E_p} t)} g(p)\bigg \Vert _{{\mathcal {F}}}^2 \bigg )^{1/2} \le { c \sum _{|\alpha |\le 2 }\sup _{p,|x|\le c_0t} \Vert \partial _{p}^{\alpha } g(p)\Vert _{{\mathcal {F}}}}, \nonumber \\ \end{aligned}$$
(4.1)
$$\begin{aligned}{} & {} \bigg ( \int _{|x|/t\ge c_0} d^3x\,\bigg \Vert \int d^3p \, e^{i(p\cdot x-E_p t)} g(p)\bigg \Vert _{{\mathcal {F}}}^2 \bigg )^{1/2} \nonumber \\{} & {} \qquad \qquad \le \, { c t^{-1/2+\varepsilon } \sum _{|\alpha |\le 2 } \sup _{p,|x|\ge c_0t} \bigg (\frac{1}{(1+t+|x|)^{\varepsilon }} \Vert \partial _{p}^{\alpha } g(p)\Vert _{{\mathcal {F}}}\bigg ).} \quad \quad \end{aligned}$$
(4.2)

The function \(g\) above may depend on (x, t).

Lemma 4.1 immediately gives the following estimate

$$\begin{aligned} \bigg ( \int d^3x\,\bigg \Vert \int d^3p{} & {} e^{i(p\cdot x- E_p t)} g(p)\bigg \Vert _{{\mathcal {F}}}^2 \bigg )^{1/2}\nonumber \\{} & {} \le c{ t^{1/2} } \sum _{|\alpha |\le 2 }\sup _{p, x} \bigg (\frac{1}{(1+|x|)^{{1/2}}} \Vert \partial _{p}^{\alpha } g(p)\Vert _{{\mathcal {F}}}\bigg ), \end{aligned}$$
(4.3)

which will be useful for analysing vectors (1.8) at finite t. Like in Lemma 4.1, the function \(g\) may depend on (x, t). We note that we cannot apply (4.3) or Lemma 4.1 directly to the infraparticle vector (1.8), since differentiability of \(p\mapsto \phi _p\) is out of control. In the course of our discussion, we will approximate \(\phi _p\) with \(\phi _{p,\sigma }\) in a suitable manner, which will introduce an x-dependence of \(g\).

As a first step of our analysis, we compute and estimate derivatives of \(e^{i \gamma (p,x,t)}\) w.r.t. p, x, t. The following is a result of a straightforward computation:

$$\begin{aligned} \partial _t e^{i \gamma (p,x,t)}= & {} e^{i \gamma (p,x,t)} {i} \int d^3k\, f_p(k)^2|k| \cos (|k|t- k\cdot x ), \end{aligned}$$
(4.4)
$$\begin{aligned} \partial _{x_i} e^{i \gamma (p,x,t)}= & {} -e^{i \gamma (p,x,t)} {i} \int d^3k\, f_p(k)^2 k_i \cos (|k|t- k\cdot x ), \end{aligned}$$
(4.5)
$$\begin{aligned} \partial _{x_j}\partial _{x_i} e^{i \gamma (p,x,t)}= & {} {-}e^{i \gamma (p,x,t)} \int d^3k\, f_p(k)^2 k_j \cos (|k|t- k\cdot x ) \nonumber \\{} & {} \times \int d^3k f_p(k)^2 k_i \cos (|k|t- k\cdot x ) \end{aligned}$$
(4.6)
$$\begin{aligned}{} & {} -\, e^{i \gamma (p,x,t)} {i} \int d^3k f_p(k)^2 k_ik_j \sin (|k|t- k\cdot x ). \end{aligned}$$
(4.7)

Now, we estimate the above expressions together with their derivatives w.r.t. p.

Lemma 4.2

The following bounds hold

$$\begin{aligned} |\partial ^{\alpha }_p \partial ^{\ell }_t e^{i \gamma (p,x,t)}|&\,\le \,&c(1+\log (1+t+|x|))^2, \end{aligned}$$
(4.8)
$$\begin{aligned} |\partial ^{\alpha }_p\partial ^{\beta }_x e^{i \gamma (p,x,t)}|&\, \le \,&c(1+\log (1+t+|x|))^2, \end{aligned}$$
(4.9)

for \(|\alpha |, |\beta | \le 2\), \(\ell \le 1\).

Proof

We see from (4.4)–(4.7) that the derivatives w.r.t. x, t produce expressions which are uniformly bounded in x, t due to the additional factors \(k_i, |k|\), which regularize the singularity of \(f_p^2\) at \(|k|=0\). Hence, it suffices to study the expression

$$\begin{aligned} \partial _{p_j}\partial _{p_i} e^{i \gamma (p,x,t)} \,= & {} \, \partial _{p_j}\big ( e^{i \gamma (p,x,t)} i \partial _{p_i}\gamma (p,x,t) \big )\nonumber \\ \,= & {} \, e^{i \gamma (p,x,t)} \big (i \partial _{p_j}\gamma (p,x,t)\big ) \big (i \partial _{p_i}\gamma (p,x,t)\big ) \nonumber \\{} & {} \quad + e^{i \gamma (p,x,t)} i \partial _{p_j}\partial _{p_i}\gamma (p,x,t). \end{aligned}$$
(4.10)

Making use of (E.4), we obtain

$$\begin{aligned} |\partial _{p_j}\partial _{p_i} e^{i \gamma (p,x,t)}|\le c(1+\log (1+t+|x|))^2, \end{aligned}$$
(4.11)

where the dependence of c on parameters is as discussed in Sect. 3. This concludes the proof. \(\square \)

As a next step of our discussion, we compute derivatives of the following auxiliary vector

$$\begin{aligned} \hat{g}_{(t,x)}(p):=W\big ( f_p m(t,x) \big ) \phi _p, \quad m(t,x):=u(t,x)-1, \quad u(t,x):=e^{-i|k| t+ik\cdot x} \nonumber \\ \end{aligned}$$
(4.12)

w.r.t. (t, x) up to the second order. We will abbreviate \(m:=m(t,x), u:=u(t,x)\).

Lemma 4.3

The function \((t,x)\mapsto {\hat{g}}_{(t,x)}(p)\) is infinitely often partially differentiable in the norm of \({\mathcal {F}}\) and the following formulas hold

$$\begin{aligned} \partial _t\hat{g}_{(t,x)}(p) \,= & {} \, W\big ( f_p m \big )i\big (\Phi ( f_p \partial _{t}m)+\textrm{Im}\langle f_p m, f_p \partial _{t} m\rangle \big ) \phi _p, \end{aligned}$$
(4.13)
$$\begin{aligned} \partial ^2_{t}\hat{g}_{(t,x)}(p)= & {} -W(f_{p}m)\big ( \Phi (f_{p} \partial _{t} m)+ \textrm{Im}\langle f_{p}m, f_p\partial _{t}m\rangle \big )^2 \phi _{p}\nonumber \\{} & {} \quad +W(f_{p}m)i\big (\Phi (f_{p} \partial _t^2 m)+\textrm{Im}\langle f_{p}m,f_{p}\partial _t^2 m\rangle \big )\phi _{p} \end{aligned}$$
(4.14)
$$\begin{aligned} \partial _{x_i}\hat{g}_{(t,x)}(p)= & {} W\big ( f_p m \big )i\big (\Phi ( f_p \partial _{x_i}m)+\textrm{Im}\langle f_p m, f_p \partial _{x_i} m\rangle \big ) \phi _p, \end{aligned}$$
(4.15)
$$\begin{aligned} \partial _{x_j}\partial _{x_i}\hat{g}_{(t,x)}(p)= & {} W\big ( f_p m \big ) i\big (\Phi ( f_p \partial _{x_j}m)+\textrm{Im}\langle f_p m, f_p \partial _{x_j} m\rangle \big ) \nonumber \\{} & {} \quad \times i\big (\Phi ( f_p \partial _{x_i}m)+\,\textrm{Im}\langle f_p m, f_p \partial _{x_i} m\rangle \big ) \phi _p\nonumber \\{} & {} \quad +\,W\big ( f_p m \big ) i\big (\Phi ( f_p \partial _{x_j}\partial _{x_i}m)+\textrm{Im}\langle f_p \partial _{x_j}m, f_p \partial _{x_i} m\rangle \nonumber \\{} & {} \quad +\, \textrm{Im}\langle f_p m, f_p \partial _{x_j}\partial _{x_i} m\rangle \big ) \phi _p, \end{aligned}$$
(4.16)

where \(\Phi (F):=a^*(-iF)+a(-iF)\), \(F\in L^2({\mathbb {R}}^3_k)\), as defined in (1.2).

Proof

We note that, by Lemma C.3, \(\phi _p\) belongs to \(D(H_{\textrm{f}}^{\ell })\) for any \(\ell \in {\mathbb {N}}\). We observe that for any fixed (t, x) the function \(f_p m(t,x)\in L^2_{\omega }({\mathbb {R}}^3_k)\), and it is infinitely often differentiable in (t, x) in the norm of \(L^2_{\omega }({\mathbb {R}}^3_k)\) (see Appendix A). For the first derivative w.r.t. \(x_i\), this follows from

$$\begin{aligned} m(t, x+(\Delta x)_i{e_i})= & {} m(t,x)+(\Delta x)_i (\partial _{x_i} m)(t,x)\nonumber \\{} & {} \quad + (\Delta x)_i^2 \int _0^{1}ds\,(1-s)\, (\partial _{x_i}^2 m)(t, x+s(\Delta x)_i{e_i}), \end{aligned}$$
(4.17)

and the fact that \(|k|^{-1}\partial _{x_i}^{\ell }m(t,x)\) is bounded in k for any \(\ell \in {\mathbb {N}}_{0}\). For higher derivatives, we simply replace m with \({\partial _{x_i}^{\ell }m}\) in (4.17). The arguments regarding the derivatives w.r.t. t are analogous. Thus, we can compute the derivatives using Lemma A.2, which gives the formulas from the statement of the lemma. \(\square \)

Now, we analyse the regularized variants of the vectors from (4.12)

$$\begin{aligned} \hat{g}^{\sigma }_{(t,x)}(p):=W\big ( f_p m(t,x) \big ) \phi _{p,\sigma }. \end{aligned}$$
(4.18)

We note the following fact:

Lemma 4.4

There hold the bounds

$$\begin{aligned} \Vert \partial ^{\alpha }_p \partial _t^{\ell } \hat{g}^{\sigma }_{(t,x)}(p)\Vert _{{\mathcal {F}}}\le & {} c \frac{(1+ \log (1+|x|+t))^{ { 3} }}{\sigma ^{\delta _{\lambda _0}}}, \end{aligned}$$
(4.19)
$$\begin{aligned} \Vert \partial ^{\alpha }_p\partial ^{\beta }_x\hat{g}^{\sigma }_{(t,x)}(p)\Vert _{{\mathcal {F}}}\le & {} c \frac{(1+\log (1+|x|+t))^{{3}} }{\sigma ^{\delta _{\lambda _0}}}, \end{aligned}$$
(4.20)

for \(\ell , |\alpha |, |\beta |\le 2\) and \(\sigma \in (0,\kappa _{\lambda _0}]\). The x and t derivatives exist in the norm of \({\mathcal {F}}\). The derivatives w.r.t. p exist in the weak sense on the domain of finite particle vectors with compactly supported wave functions (cf. [34, p. 208]). The bound (4.20) still holds if \(\partial ^{\beta }_x\) is replaced with \(H_{\textrm{f}}, P_{\textrm{f},i}, P_{\textrm{f},i}^2\) or \(\partial _{x_i}P_{\textrm{f},i}\).

Proof

We consider only (4.20) for \(|\alpha |=2, |\beta |=2\) as the remaining cases are analogous and simpler. To handle the resulting expressions, it is convenient to define, for \(s\mapsto F_s\) as in Lemma A.2,

$$\begin{aligned} \widetilde{\Phi }_s(F):=\Phi (\partial _s F_s)+\textrm{Im}\langle F_s, \partial _s F_s\rangle . \end{aligned}$$
(4.21)

Using this notation and recalling (4.16), we can write

$$\begin{aligned} \partial _{x_j}\partial _{x_i}\hat{g}^{\sigma }_{(t,x)}(p)= & {} W\big ( f_p m \big ) \bigg \{ i\widetilde{\Phi }_{x_j}( f_p m) i\widetilde{\Phi }_{x_i}( f_p m) + i \partial _{x_j} \widetilde{\Phi }_{x_i}( f_p m)\bigg \} \phi _{p,\sigma }\nonumber \\= & {} W\big ( f_p m \big ){ \textrm{Pol}}_{x_i,x_j}(f_pm) \phi _{p,\sigma }, \end{aligned}$$
(4.22)

where in the last step we denoted the expression in curly brackets by the symbol \(\textrm{Pol}_{x_i,x_j}(f_{p}m)\) to further abbreviate the notation. Now, we compute the first derivative w.r.t. momentum. We recall that these derivatives must only exist weakly on the domain of finite particle vectors, i.e. after taking a scalar product with such vectors. This will control the unbounded operators acting on \(\phi _{p,\sigma }\) below and, in particular, allow us to differentiate \(p\mapsto \phi _{p,\sigma }\) in (4.25) below. In this sense, we compute:

$$\begin{aligned} \partial _{p_{{\hat{i}}}} \partial _{x_j}\partial _{x_i}\hat{g}^{\sigma }_{(t,x)}(p)= & {} W\big ( f_p m \big )i \widetilde{\Phi }_{ p_{{\hat{i}}} }(f_p m) { \textrm{Pol}}_{x_i,x_j}(f_pm) \phi _{p,\sigma } \end{aligned}$$
(4.23)
$$\begin{aligned}{} & {} +\, W\big ( f_p m \big ) \partial _{p_{{\hat{i}}}} \big ( { \textrm{Pol}}_{x_i,x_j}(f_pm)\big ) \phi _{p,\sigma } \end{aligned}$$
(4.24)
$$\begin{aligned}{} & {} +\, W\big ( f_p m \big ) {\textrm{Pol}}_{x_i,x_j}(f_pm) \partial _{p_{{\hat{i}}}}\phi _{p,\sigma }. \end{aligned}$$
(4.25)

Now, we compute the respective contributions to \(\partial _{p_{{\hat{j}}}} \partial _{p_{{\hat{i}}}} \partial _{x_j}\partial _{x_i}{\hat{g}}^{\sigma }_{(t,x)}(p)\): (4.23) gives

$$\begin{aligned}{} & {} \partial _{p_{{\hat{j}}}} \big (W\big ( f_p m \big )i \widetilde{\Phi }_{p_{{\hat{i}}}}(f_p m) { \textrm{Pol}}_{x_i,x_j}(f_pm) \phi _{p,\sigma }\big )\nonumber \\{} & {} \quad = W\big ( f_p m \big )i \widetilde{\Phi }_{p_{{\hat{j}}}}(f_p m) i \widetilde{\Phi }_{p_{{\hat{i}}}}(f_p m) { \textrm{Pol}}_{x_i,x_j}(f_pm) \phi _{p,\sigma } \end{aligned}$$
(4.26)
$$\begin{aligned}{} & {} \qquad +\,W\big ( f_p m \big )i \partial _{p_{{\hat{j}}}} (\widetilde{\Phi }_{p_{\hat{i}}}(f_p m) {\textrm{Pol}}_{x_i,x_j}(f_pm)) \phi _{p,\sigma } \end{aligned}$$
(4.27)
$$\begin{aligned}{} & {} \qquad +\,W\big ( f_p m \big )i \widetilde{\Phi }_{p_{\hat{i}}}(f_p m) { \textrm{Pol}}_{x_i,x_j}(f_pm) \partial _{p_{{\hat{j}}}}\phi _{p,\sigma }. \end{aligned}$$
(4.28)

From (4.24), we obtain

$$\begin{aligned} \partial _{p_{{\hat{j}}}} \big (W\big ( f_p m \big ) \partial _{p_{{\hat{i}}}} { \textrm{Pol}}_{x_i,x_j}(f_pm) \phi _{p,\sigma }\big )= & {} W\big ( f_p m \big ) i\widetilde{\Phi }_{p_{{\hat{i}}}}(f_p m) \partial _{p_{{\hat{i}}}} { \textrm{Pol}}_{x_i,x_j}(f_pm) \phi _{p,\sigma }\nonumber \\{} & {} +\, W\big ( f_p m \big ) \partial _{p_{{\hat{j}}}}\partial _{p_{{\hat{i}}}} \big ({ \textrm{Pol}}_{x_i,x_j}(f_pm)\big ) \phi _{p,\sigma } \nonumber \\{} & {} +\,W\big ( f_p m \big ) \partial _{p_{{\hat{i}}}} { \textrm{Pol}}_{x_i,x_j}(f_pm) \partial _{p_{{\hat{j}}}} \phi _{p,\sigma }.\nonumber \\ \end{aligned}$$
(4.29)

From (4.25), we get

$$\begin{aligned} \partial _{p_{{\hat{j}}}} \big ( W\big ( f_p m \big ) { \textrm{Pol}}_{x_i,x_j}(f_pm) \partial _{p_{{\hat{i}}}}\phi _{p,\sigma }\big )= & {} W\big ( f_p m \big ) i\widetilde{\Phi }_{p_{{\hat{i}}}}(f_p m) \ { \textrm{Pol}}_{x_i,x_j}(f_pm) \partial _{p_{{\hat{i}}}} \phi _{p,\sigma }\nonumber \\{} & {} +\, W\big ( f_p m \big ) \partial _{p_{{\hat{j}}}} \big ({ \textrm{Pol}}_{x_i,x_j}(f_pm)\big ) \partial _{p_{{\hat{i}}}}\phi _{p,\sigma } \nonumber \\{} & {} +\, W\big ( f_p m \big ) { \textrm{Pol}}_{x_i,x_j}(f_pm) \partial _{p_{{\hat{i}}}}\partial _{p_{{\hat{j}}}} \phi _{p,\sigma }.\nonumber \\ \end{aligned}$$
(4.30)

To estimate these expressions, we recall from Lemma 3.2 that \(\partial ^{\alpha }_p \phi _{p,\sigma }\) are in the domain of any power of N and \(\Vert N^{\ell } \partial ^{\alpha }_p \phi _{p,\sigma }\Vert _{{\mathcal {F}}} \le c_{\ell } \sigma ^{-\delta _{\lambda _0}}\). Thus making use of the number bounds (A.2), we have

$$\begin{aligned} \Vert \partial _{p_{{\hat{j}}}} \partial _{p_{{\hat{i}}}} \partial _{x_j}\partial _{x_i}\hat{g}^{\sigma }_{(t,x)}(p)\Vert _{{\mathcal {F}}}\le { \textrm{Pol}}( \Vert f_pm\Vert _2, \Vert f_p\partial _{x_i}m\Vert _2, \Vert f_p \partial _{x_j}\partial _{x_i}m\Vert _2) \sigma ^{-\delta _{\lambda _0}}.\qquad \end{aligned}$$
(4.31)

Here \({ \textrm{Pol}}\) is a certain polynomial in the specified norms, which also includes \(\Vert \partial _{p}^{\alpha }{ f_p} m\Vert _2\). We recall, however, that \(f_p(k):=\lambda \frac{\chi _{\kappa }(k)}{\sqrt{2|k|}}\frac{1}{|k|(1-e_k\cdot \nabla E_{p} )}\), thus derivatives of \(f_p\) w.r.t. p only change the behaviour of this function in the angular variable \(e_k\) but not in the |k|-variable. As our estimates are insensitive to the angular behaviour, we omitted these derivatives in the notation in (4.31). We have

$$\begin{aligned} \Vert f_pm\Vert _2\le c|\lambda |(1+\log (1+|x|+t))^{1/2}, \, \Vert f_p\partial _{x_i}m\Vert _2\le c|\lambda |,\,\, \Vert f_p\partial _{x_i}\partial _{x_j}m\Vert _2\le c|\lambda |, \nonumber \\ \end{aligned}$$
(4.32)

where the first inequality follows from Lemma E.3 and the last two follow directly from the definition of \(f_p\) in (1.7), since \(m(t,x):=e^{-i|k|t+ik\cdot x}-1\). By inspection, we see that \({ \textrm{Pol}}\) is at most of the sixth order in \(\Vert f_pm\Vert _2\) (cf. (4.26)), which concludes the proof of estimates (4.19), (4.20).

As for the last statement of the lemma, the case of \(H_{\textrm{f}}, P_{\textrm{f},i}, P_{\textrm{f},i}^2\) is covered by the fact that the derivatives w.r.t. p should exist only weakly on vectors which belong to domains of these operators. After computing these derivatives, one pulls \(H_{\textrm{f}}, P_{\textrm{f},i}, P_{\textrm{f},i}^2\) to the right through the Weyl operator according to

$$\begin{aligned} H_{\textrm{f}}W(f_{p}m)= W(f_{p}m)(H_{\textrm{f}}+a^*({|k|}f_pm)+a({|k|}f_pm) +\Vert {|k|^{1/2}} f_p m \Vert _2^2), \nonumber \\ \end{aligned}$$
(4.33)

for which we refer to [12, eq. (3.20) and Prop. 3.11]. Next one applies Lemmas 3.2 and A.1. The case of \(P_{\textrm{f},i} \partial _{x_i}\) requires more consideration as the derivative w.r.t. \(x_i\) should exist in the norm of \({\mathcal {F}}\). To check that \(P_{\textrm{f},i}W(f_{p}m)\phi _{p,\sigma }\) is partially differentiable w.r.t. x in the norm of \({\mathcal {F}}\), we write, analogously to (4.33),

$$\begin{aligned} P_{\textrm{f},i}W(f_{p}m)\phi _{p,\sigma }= W(f_{p}m)(P_{\textrm{f},i}+a^*(k_if_pm)+a(k_i f_pm ) +\langle f_p, k_i f_p\rangle ) \phi _{p,\sigma } \nonumber \\ \end{aligned}$$
(4.34)

and refer to Lemma A.2. By a computation, we obtain

$$\begin{aligned} \partial _{x_i} P_{\textrm{f},i}W(f_{p}m )\phi _{p,\sigma }= P_{\textrm{f},i} \partial _{x_i}\big ( W(f_{p}m)\big )\phi _{p,\sigma }, \end{aligned}$$
(4.35)

where \(\partial _{x_i}\big ( W(f_{p}m)\big )\) is the explicit formula from Lemma A.2, and then proceed as in the discussion of \(H_{\textrm{f}}\), \(P_{\textrm{f},i}\), \(P_{\textrm{f},i}^2\) above. \(\square \)

Now, we are ready to analyse the infraparticle vector (1.8).

Lemma 4.5

There is such \(\lambda _0>0\) that for any \(|\lambda |\in (0,\lambda _0]\) and \(t\in {\mathbb {R}}\), the integralFootnote 3

$$\begin{aligned} \Psi _t(x):=\int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m(t,x) \big ) \phi _p \end{aligned}$$
(4.36)

has the following properties:

  1. (a)

    \(\Psi _t\in L^2({\mathbb {R}}^3_x;{\mathcal {F}})\).

  2. (b)

    \(\Psi _t\) is differentiable in t in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) and

    $$\begin{aligned} \partial _t\Psi _t(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } \big (-iE_p+i\partial _t \gamma (p,x,t) +i\textrm{Im}\langle f_pm, f_p\partial _t m\rangle \big ) \nonumber \\{} & {} \quad \times ~e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \phi _p\nonumber \\{} & {} \quad + \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big )(a^*(f_p\partial _t m)-a(f_p\partial _tm)) \phi _p.\nonumber \\ \end{aligned}$$
    (4.37)

Proof

As for (a), to prove that \(x\mapsto \Psi _t(x)\) is square integrable, we intend to apply Lemma 4.1. However, we lack information about the differentiability of \(p\mapsto \phi _p\). To circumvent this problem, we introduce an x-dependent cut-off \(\sigma _{x}:=\kappa _{\lambda _0}/(1+|x|)^M\), where M is sufficiently large but fixed. We insert into (4.36)

$$\begin{aligned} \phi _p=(\phi _p-\phi _{p,\sigma _{x}})+\phi _{p,\sigma _{x}} \end{aligned}$$
(4.38)

and obtain

$$\begin{aligned} \Psi _t(x)=\int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( {f_p m} \big ) (\phi _p - \phi _{p,\sigma _{x}})+\Psi _t^{\sigma _{x}}(x). \nonumber \\ \end{aligned}$$
(4.39)

Here, \(\Psi _t^{\sigma _{x}}(x)\) is given by (4.36) with \(\phi _p\) replaced with \(\phi _{p,\sigma _{x}}\). Concerning the first term on the r.h.s. of (4.39), we have by (3.5)

$$\begin{aligned} \bigg \Vert \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( {f_p m} \big ) (\phi _p - \phi _{p,\sigma _{{x}}})\bigg \Vert _{{\mathcal {F}}}\le \frac{c (\kappa _{\lambda _0})^{ 1/5 } }{(1+|x|)^{{M}/5} }.\nonumber \\ \end{aligned}$$
(4.40)

Thus, this term is manifestly in \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) for \(2M/5>3\). As for the last term on the r.h.s. of (4.39), estimate (4.3) gives

$$\begin{aligned} \Vert \Psi _t^{\sigma _{x}}\Vert _{{\mathcal {H}}}\le c {t^{1/2} } \sum _{|\alpha |\le 2 } \sup _{p,x} \bigg (\frac{1}{(1+|x|)^{{1/2}}} \Vert \partial _{p}^{\alpha } (e^{i \gamma (p,x,t)} \hat{{g}}^{\sigma _x}_{(t,x)}(p))\Vert _{{\mathcal {F}}}\bigg ). \end{aligned}$$
(4.41)

The expression on the r.h.s. above is finite for any fixed t by Lemmas 4.2, 4.4, provided \(\delta _{\lambda _0}\) of Lemma 4.4 satisfies \(M\delta _{\lambda _0}<1/2\). This concludes the proof of part (a).

Part (b) is a straightforward computation, provided we can show differentiability in the norm of \(L^2({\mathbb {R}}^3_x; {\mathcal {F}})\). To this end, we use the Taylor theorem (cf. formula (4.17))

$$\begin{aligned}{} & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p)\nonumber \\{} & {} \qquad \qquad \quad \quad \times \bigg ( \frac{W\big ( f_p m(t+\Delta t,x) \big ) - W\big ( f_p m(t,x) \big )}{\Delta t} -\partial _{t}W\big ( f_p m(t,x) \big ) \bigg ) \phi _p\nonumber \\{} & {} \qquad \qquad = \Delta t \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p)\nonumber \\{} & {} \qquad \qquad \quad \quad \times \int _0^1 ds\,(1-s) {\big \{\partial _{\tau }^2W\big ( f_p m(\tau ,x) \big )|_{\tau =t+s\Delta t }\big \}} \phi _p. \end{aligned}$$
(4.42)

Now, we obtain from Lemma A.3 that (4.42) tends to zero with \(\Delta t\rightarrow 0\) in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\). Differentiability in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) of other ingredients of (4.36) can be shown by analogous and simpler arguments. Now, formula (4.37) follows by an application of Lemma A.2. \(\square \)

Lemma 4.6

The vectors \(\Psi _t\in L^2({\mathbb {R}}^3_x;{\mathcal {F}})\), \(t\in {\mathbb {R}}\), defined in (4.36) have the following properties:

  1. (a)

    \(\Psi _t\) is in the domain of \(P_{\textrm{f},i}, P_{\textrm{f},i}^2\), \(H_{\textrm{f}}\) and the following formula holds

    $$\begin{aligned} (H_{\textrm{f}} \Psi _t)(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big )\nonumber \\{} & {} \quad \times \big (H_{\textrm{f}}^{\textrm{w}}+a^*(|k|f_p u)+a(|k|f_pu)\nonumber \\{} & {} \qquad +{\langle f_p,|k| f_p\rangle } -2\textrm{Re}\langle f_p, |k| f_p u\rangle \big ) \phi _p. \end{aligned}$$
    (4.43)
  2. (b)

    \(\Psi _t\) is in the domain of \(-i\partial _{x_i}\), \((-i\partial _{x_i})^2\), \(-i\partial _{x_i}P_{\textrm{f},i}\) and the following formula holds

    $$\begin{aligned} (-i\partial _{x_i}- P_{\textrm{f},i})^{2} \Psi _t(x)= \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \big (p_i-P_{\textrm{f},i}^{\textrm{w}} \big )^{2} \phi _p.\nonumber \\ \end{aligned}$$
    (4.44)
  3. (c)

    \(\Psi _t\) is in the domain of \((a^*(v)+a(v))\) and the following formula holds

    $$\begin{aligned} (a^*(v)+a(v)) \Psi _t(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \nonumber \\{} & {} \quad \times ((a^*(v)+a(v))^{\textrm{w}}+ \,2\textrm{Re}\langle f_pu, v\rangle ) \phi _p, \end{aligned}$$
    (4.45)

    where \((a^*(v)+a(v))^{\textrm{w}}=a^*(v)+a(v)-2\langle f_p,v\rangle \) in accordance with (1.7).

Proof

We start with some computations on \({\mathcal {F}}\) which are justified by Lemma A.2. Since \(W\big ( f_p m \big )\phi _p\) is in the domain of \(H_{\textrm{f}}\), we can write for any fixed t

$$\begin{aligned} H_{\textrm{f}}\Psi _t(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) H_{\textrm{f}}W\big ( f_p m \big ) \phi _p \nonumber \\= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big )\nonumber \\{} & {} \quad \times \big (H_{\textrm{f}} {+}a^*(|k|f_p m){+}a(|k|f_pm)+\Vert |k|^{1/2} f_pm\Vert _2^2\big ) \phi _p\nonumber \\= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big )\nonumber \\{} & {} \quad \times \big (H_{\textrm{f}}^{\textrm{w}} +a^*(|k|f_p u)+a(|k|f_pu) \nonumber \\{} & {} \qquad +\langle f_p,|k| f_p\rangle -2\textrm{Re}\langle f_p, |k| f_p u\rangle \big ) \phi _p, \,\, \end{aligned}$$
(4.46)

where we made use of \(H_{\textrm{f}}^{\textrm{w}}=H_{\textrm{f}}-a^*(|k|f_p)-a(|k|f_p)+\Vert \, |k|^{1/2}f_p\Vert _2^2\) (cf. formula (4.33)) and

$$\begin{aligned} -\Vert |k|^{1/2} f_p \Vert _2^2 +\Vert |k|^{1/2} f_pm\Vert _2^2= \langle f_p,|k| f_p\rangle -2\textrm{Re}\langle f_p, |k| f_p u\rangle . \end{aligned}$$
(4.47)

Analogously, we obtain for \({\ell \in \{1,2\} }\),

$$\begin{aligned} (P_{\textrm{f},i})^{\ell } \Psi _t(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \nonumber \\{} & {} \quad \!\times \!\big (P^{\textrm{w}}_{\textrm{f},i}+a^*(k_if_p {u}) \!+\!a(k_if_pu)\!+\! \langle f_p,k_i f_p\rangle -\,2\textrm{Re}\langle f_p, k_i f_p u \rangle \big )^{\ell } \phi _p.\nonumber \\ \end{aligned}$$
(4.48)

Furthermore, we can exchange \(-i\partial _{x_i}\) with the p-integral defining \(\Psi _t\). In fact, similarly as in (4.42), we write

$$\begin{aligned}{} & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) \bigg ( \frac{W\big ( f_p m(t,x+ (\Delta x_i)e_i ) \big ) - W\big ( f_p m(t,x) \big )}{\Delta x_i} \nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \quad -\partial _{x_i}W\big ( f_p m(t,x) \big ) \bigg ) \phi _p\nonumber \\{} & {} \quad = (\Delta x_i) \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p)\nonumber \\{} & {} \qquad \times \int _0^1 ds\,(1-s) {\big \{\partial _{x'_i}^2W\big ( f_{p}m(t,x') \big )|_{x'=x+s(\Delta x_i)e_i }\big \}} \phi _p \end{aligned}$$
(4.49)

and make use of Lemma A.3 to take the limit \(\Delta x_i \rightarrow 0\). Thus, we can write

$$\begin{aligned} -i\partial _{x_i}\Psi _t(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p)\nonumber \\{} & {} \quad \times \big (p_i+\partial _{x_i}\gamma (p,x,t)+\textrm{Im}\langle f_{p}m,f_{p} \partial _{x_i}m \rangle \big )W(f_{p}m)\phi _p \nonumber \\{} & {} \quad + \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) \nonumber \\{} & {} \quad \times (a^*(k_{i}f_{p}u)+ a(k_{i}f_{p}u))\phi _p. \end{aligned}$$
(4.50)

Combining the above computations, we also obtain

$$\begin{aligned}{} & {} -i\partial _{x_i}P_{\textrm{f},i} \Psi _t(x) \nonumber \\{} & {} \quad = \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) \big (p_i+\partial _{x_i} \gamma (p,x,t)+\textrm{Im}\langle f_pm, f_p\partial _{x_i} m\rangle \big ) \nonumber \\{} & {} \qquad \times W\big ( f_p m \big )\big (P_{\textrm{f},i}{+}a^*(k_if_p m) {+}a(k_if_pm) +\langle f_pm, k_i f_pm \rangle \big ) \phi _p\nonumber \\{} & {} \qquad + \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p)W\big ( f_p m \big )\nonumber \\{} & {} \qquad \times \big (P_{\textrm{f},i}{+}a^*(k_if_p m) {+}a(k_if_pm)+\langle f_pm, k_i f_pm \rangle \big ) \nonumber \\{} & {} \qquad \times (a^*( k_i f_p u)+a( k_i f_p u)) \phi _p=P_{\textrm{f},i}(-i\partial _{x_i}) \Psi _t(x). \end{aligned}$$
(4.51)

Thus, we get from (4.48) and (4.50)

$$\begin{aligned}{} & {} (-i\partial _{x_i}- P_{\textrm{f},i}) \Psi _t(x)= \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \nonumber \\{} & {} \qquad \times \big (-P_{\textrm{f},i}^{\textrm{w}} +p_i+\partial _{x_i} \gamma (p,x,t)+ \textrm{Im}\langle f_pm, f_p\partial _{x_i} m\rangle - \langle f_p,k_i f_p\rangle \nonumber \\{} & {} \qquad \quad +2\textrm{Re}\langle f_p, k_i f_p u \rangle \big ) \phi _p\nonumber \\{} & {} \quad = \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \big (p_i-P_{\textrm{f},i}^{\textrm{w}}) \phi _p, \end{aligned}$$
(4.52)

where we used that

$$\begin{aligned}{} & {} \textrm{Im}\langle f_pm, f_p\partial _{x_i} m\rangle - \langle f_p,k_i f_p\rangle + 2\textrm{Re}\langle f_p, k_i f_p u \rangle \nonumber \\{} & {} =\textrm{Re}\langle f_p , k_i f_pu\rangle = -\partial _{x_i} \gamma (p,x,t). \end{aligned}$$
(4.53)

By iteration of (4.52), we get

$$\begin{aligned} (-i\partial _{x_i}- P_{\textrm{f},i})^{\ell } \Psi _t(x)= \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) \big (p_i-P_{\textrm{f},i}^{\textrm{w}} \big )^{\ell } \phi _p. \nonumber \\ \end{aligned}$$
(4.54)

We remark that at the level of formal computations, relations (4.52), (4.54) can also be obtained from (1.12). Finally, we obtain

$$\begin{aligned} (a^*(v)+a(v)) \Psi _t(x)= & {} \int d^3p \, e^{i(p\cdot x-E_pt) } e^{i \gamma (p,x,t)} h(p) W\big ( f_p m \big ) ( (a^*(v)+a(v))^{\textrm{w}}\nonumber \\{} & {} \quad +\, 2\textrm{Re}\langle f_pu, v\rangle ) \phi _p. \end{aligned}$$
(4.55)

One can see, by analogous arguments as in the proof of Lemma 4.5 (a), that all vectors above are in \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\): First, we apply the shift (4.39) and estimate the term involving \(\phi _p-\phi _{p,\sigma _x}\) with the help of the bound (3.5). The presence of \(H_{\textrm{f}}^{\ell }\) in (3.5) allows us to control both \(P_{\textrm{f},i}\) and the creation and annihilation operators acting on \(\phi _p-\phi _{p,\sigma _x}\) as for example in the case of (4.51). To the latter operators, we apply the energy bounds (A.3) and note that all the resulting \(\Vert \,\cdot \,\Vert _{\omega }\)-norms are finite. Next, we study the term proportional to \(\phi _{p,\sigma _x}\) using Lemma 4.1. Staying with the case of (4.51), we can rewrite the relevant vector as \({ \{ P_{\textrm{f},i}(-i\partial _{x'_i}\Psi ^{\sigma _x}_t(x') ) |_{x'=x}\}_{x\in {\mathbb {R}}^3} }\) and estimate the r.h.s. of (4.3) using Lemmas 4.2, 4.4. In particular, the last part of Lemma 4.4 plays a role here, since estimate (4.3) gives

$$\begin{aligned}{} & {} \Vert \{P_{\textrm{f},i}(-i\partial _{x'_i}\Psi ^{\sigma _x}_t(x') ) |_{x'=x}\}_{x\in {\mathbb {R}}^3} \Vert _{{\mathcal {H}}} \nonumber \\{} & {} \quad \le c {t^{1/2} }\sum _{|\alpha |\le 2 } \sup _{p,x} \bigg (\frac{1}{(1+|x|)^{{1/2}}} \Vert \partial _{p}^{\alpha } ( \{\partial _{x'_i} e^{i \gamma (p,x',t)}P_{\textrm{f},i}\hat{{g}}^{\sigma _x}_{(t,x')}(p)\}|_{x'=x} )\Vert _{{\mathcal {F}}}\bigg ). \nonumber \\ \end{aligned}$$
(4.56)

From (4.54), (4.51) we also obtain that \({\{ (-i\partial _{x_i})^2\Psi _t(x)\}_{x\in {\mathbb {R}}^3}}\) is in \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\). This concludes the proof. \(\square \)

Proof of Theorem 1.1

We recall that as seen in  (1.8). By Lemma 4.5, \(t\mapsto \Psi _t\) is differentiable in the norm in \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\). Next, by applying the Stone theorem to \(e^{iHt}\), we obtain the differentiability of \( t\mapsto \psi _t\) in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\), provided that the vector \(\{e^{{-}iP_{\textrm{f}}\cdot x} \Psi _t(x)\}_{x\in {\mathbb {R}}^3}\in L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) is in the domain of H. This is easily checked using Lemma 4.6. In particular, to verify that this vector is in the domain of \((-i\nabla _x)^2\), we apply the Stone theorem to \(x\mapsto e^{{-}iP_{\textrm{f}}\cdot x}\) and use that \(\Psi _t\) is in the domain of \(P_{\textrm{f}}^2\). Now, we compute

$$\begin{aligned} \partial _t\psi _t(x)= & {} \frac{1}{(2\pi )^{3/2}} e^{iHt} iHe^{ {-}iP_{\textrm{f}}\cdot x} \Psi _t(x)+\frac{1}{(2\pi )^{3/2}} e^{iHt} e^{{-}iP_{\textrm{f}}\cdot x} \partial _t\Psi _t(x)\nonumber \\= & {} \frac{1}{(2\pi )^{3/2}}e^{iHt} e^{{-}iP_{\textrm{f}}\cdot x} i\bigg ( \frac{1}{2}(-i\nabla _x-P_{\textrm{f}})^2 \Psi _t(x)+ H_{\textrm{f}} \Psi _t(x)\nonumber \\{} & {} \quad + (a^*(v)+a(v))\Psi _t(x)-i\partial _t\Psi _t(x)\bigg )\nonumber \\= & {} \frac{1}{(2\pi )^{3/2}}e^{iHt} e^{-iP_{\textrm{f}}\cdot x } \int d^3p \, e^{i (p \cdot x -E_pt )} e^{i \gamma (p,x,t)} i \gamma _{\textrm{int}}(p,x,t) h(p) W(f_p m) \phi _p,\nonumber \\ \end{aligned}$$
(4.57)

where in the last step we made use of the formulas in Lemmas 4.5, 4.6, the fact that \(H^{\textrm{w}}_p\phi _p=E_p\phi _p\), and of the relations

$$\begin{aligned} \begin{aligned}&\langle f_p,|k| f_p\rangle -2\textrm{Re}\langle f_p,|k| f_pu\rangle +\partial _t\gamma (p,x,t)+\textrm{Im}\langle f_pm,f_p\partial _tm\rangle =0,\\&\quad 2\textrm{Re}\langle f_pu(t,x),v\rangle = \gamma _{\textrm{int}}(p,x,t),\ \end{aligned} \end{aligned}$$
(4.58)

where v was defined below (1.3). To show (1.11), we proceed similarly as in the proof of Lemma 4.5: We choose a (t, x)-dependent cut-off as follows: \(\sigma _{(t,x)}=\kappa _{\lambda _0}/(1+t+|x|)^M\) where \(M\in {\mathbb {N}}\) is fixed. We make a shift \(\phi _p=(\phi _p-\phi _{p,\sigma _{(t,x)}})+\phi _{p,\sigma _{(t,x)}}\) and insert it into the formula for the norm of \(\partial _t\psi _t\):

$$\begin{aligned}{} & {} \Vert \partial _t\psi _t\Vert _{{\mathcal {H}}} \le \frac{1}{(2\pi )^{3/2}} \bigg \Vert \bigg \{ \int d^3p \, e^{i (p \cdot x -E_pt )} e^{i \gamma (p,x,t)} i \gamma _{\textrm{int}}(p,x,t) h(p)\nonumber \\{} & {} \qquad \qquad \times W(f_p (e^{-i|k| t+ik\cdot x }-1)) (\phi _p - \phi _{p,\sigma _{(t,x)}}) \bigg \}_{x\in {\mathbb {R}}^3} \bigg \Vert _{{\mathcal {H}}} \nonumber \\{} & {} \quad + \frac{1}{(2\pi )^{3/2}}\bigg \Vert \bigg \{\int d^3p \, e^{i (p \cdot x -E_pt )} e^{i \gamma (p,x,t)}i \gamma _{\textrm{int}}(p,x,t) h(p)\nonumber \\{} & {} \qquad \qquad \times W(f_p (e^{-i|k| t+ik\cdot x }-1)) \phi _{p,\sigma _{(t,x)}} \bigg \}_{x\in {\mathbb {R}}^3} \bigg \Vert _{{\mathcal {H}}}. \end{aligned}$$
(4.59)

We note that by (3.5) the term involving \((\phi _p-\phi _{p,\sigma _{(t,x)}}) \) is integrable in t in the norm of \(L^2({\mathbb {R}}^3_x;{\mathcal {F}})\) for M sufficiently large. Our strategy to estimate the second term on the r.h.s. of (4.59) is to combine Lemmas 4.1, 4.7 and 3.2. In our case, \(g\) of Lemma 4.1 has the form

$$\begin{aligned} g_{(t,x)}(p):=e^{i \gamma (p,x,t)} i \gamma _{\textrm{int}}(p,x,t) h(p)W(f_p (e^{-i|k| t+ik\cdot x }-1)) \phi _{p,\sigma _{(t,x)}}. \end{aligned}$$
(4.60)

We rewrite this expression as follows:

$$\begin{aligned} \begin{aligned} g_{(t,x)}(p)=&\,\,e^{i \gamma (p,x,t)} i \gamma _{\textrm{int}}(p,x,t) h(p) \hat{g}^{\sigma _{(t,x)}}_{(t,x)}{ (p)}, \\ \hat{g}^{\sigma }_{(t,x)}(p):=&\,\,W(f_p (e^{-i|k| t+ik\cdot x }-1) )\phi _{p,\sigma }. \end{aligned} \end{aligned}$$
(4.61)

First, we note that by Lemma 4.7, for \(c_0\) as in Lemma 4.1,

$$\begin{aligned}&|\partial _{p}^{\alpha } \gamma _{\textrm{int}}(p,x,t)|\le |\lambda |^2\frac{c_{\tilde{M}}}{t^{\tilde{M}}} \quad \text { for }\quad |x|/t\le c_0<1, \end{aligned}$$
(4.62)
$$\begin{aligned}&|\partial _{p}^{\alpha } \gamma _{\textrm{int}}(p,x,t)|\le |\lambda |^2\frac{c}{t} |\log \,(t)| \quad \text { for }\quad |x|/t\ge c_0, \end{aligned}$$
(4.63)

and \(|\alpha |=0,1,2\). Furthermore, we have by Lemma 4.2

$$\begin{aligned}{} & {} |\partial _p^{\alpha } e^{i \gamma (p,x,t)}|\le c (1+\log (1+t+|x|))^2. \end{aligned}$$
(4.64)

Given (4.62)–(4.64), Lemmas 4.1 and 4.4, for any \(0<\varepsilon <1/2\), we can choose \(\lambda _0\) so small, that

$$\begin{aligned} \Vert \partial _t\psi _t\Vert _{{\mathcal {H}}}\le |\lambda |^2\frac{c}{t^{3/2-\varepsilon }} \end{aligned}$$
(4.65)

which concludes the proof of (1.11). Hence, by the Cook method [10], we obtain the existence of the limit \(\psi ^{+}\).

To see that \(\psi ^+\ne 0\) under the specified conditions, we write

$$\begin{aligned} \Vert \psi ^{+,(\lambda )}\Vert _{{\mathcal {H}}} \ge \Vert \psi _{t=0}^{(\lambda )}\Vert _{{\mathcal {H}}}- \int _0^{\infty } \hbox {d}t\, \Vert \partial _t\psi ^{(\lambda )}_t\Vert _{{\mathcal {H}}}, \end{aligned}$$
(4.66)

where we included the dependence on \(\lambda \) explicitly in the notation. We recall that all constants in our discussion are uniformly bounded in \(|\lambda |\in (0,\lambda _0]\). Thus, by estimate (4.65), the second term on the r.h.s. of (4.66) tends to zero as \(\lambda \rightarrow 0\). So it suffices to show that \(\Vert \psi _{t=0}^{(\lambda )}\Vert _{{\mathcal {H}}}\) is bounded from below uniformly in \(\lambda \) from some neighbourhood of zero. We collect the relevant ingredients: First, we recall that by [15, formula (5.2)]

$$\begin{aligned} \Vert \phi ^{(\lambda )}_p-\Omega \Vert _{{\mathcal {F}}}\le c|\lambda |^{1/4}. \end{aligned}$$
(4.67)

Furthermore, we obtain from (4.32), (E.4)

$$\begin{aligned} \begin{aligned} \Vert f^{(\lambda )}_p (e^{-i|k|t+ik\cdot x }-1)\Vert _2 \le&c|\lambda |(1+\log (1+t+|x|))^{1/2},\\ | \gamma (p,x,t)|\le&c|\lambda |^{2}(1+\log (1+t+|x|)). \end{aligned} \end{aligned}$$
(4.68)

Considering the above, we have

$$\begin{aligned}{} & {} \psi _{t=0}^{(\lambda )}(x)\nonumber \\{} & {} \quad = \frac{1}{(2\pi )^{3/2}} \int d^3p \, e^{i p \cdot x } e^{i \gamma ^{(\lambda )}(p,x,0)} h(p) W(f^{(\lambda )}_p (e^{ik\cdot x }-1)) \phi ^{(\lambda )}_p \end{aligned}$$
(4.69)
$$\begin{aligned}{} & {} \quad = \frac{1}{(2\pi )^{3/2}}\int d^3p \, e^{i p \cdot x } e^{i \gamma ^{(\lambda )}(p,x,0)} h(p) W(f^{(\lambda )}_p (e^{ik\cdot x }-1)) (\phi ^{(\lambda )}_p-\Omega )\qquad \qquad \end{aligned}$$
(4.70)
$$\begin{aligned}{} & {} \qquad +\frac{1}{(2\pi )^{3/2}}\int d^3p \, e^{i p \cdot x } e^{i \gamma ^{(\lambda )}(p,x,0)} h(p) \big (W(f^{(\lambda )}_p (e^{ik\cdot x }-1)) - 1\big ) \Omega \end{aligned}$$
(4.71)
$$\begin{aligned}{} & {} \qquad +\frac{1}{(2\pi )^{3/2}}\int d^3p \, e^{i p \cdot x } \big (e^{i \gamma ^{(\lambda )}(p,x,0)}-1\big ) h(p) \Omega \end{aligned}$$
(4.72)
$$\begin{aligned}{} & {} \qquad +\frac{1}{(2\pi )^{3/2}}\int d^3p \, e^{i p \cdot x } h(p) \Omega . \end{aligned}$$
(4.73)

Thus, it is manifest from estimates (4.68), (4.67), combined with an argument as in (A.9) that

$$\begin{aligned} \psi _{t=0}^{(\lambda )}(x)=({F^{-1}} h)(x)\Omega +O(|\lambda |^{1/4} (1+\log (1+|x|)) ), \end{aligned}$$
(4.74)

where F is the Fourier transform and we have \(\Vert O(|\lambda |^{1/4} (1+\log (1+|x|)) )\Vert _{{\mathcal {F}}}\le c|\lambda |^{1/4} (1+\log (1+|x|))\). Clearly, we can write for any compact subset \(\Delta \subset {\mathbb {R}}^3\)

$$\begin{aligned} \Vert \psi _{t=0}^{(\lambda )}\Vert _{{\mathcal {H}}}\ge & {} \bigg ( \int _{\Delta } d^3x\, \Vert \psi _{t=0}^{(\lambda )}(x)\Vert ^2_{{\mathcal {F}}} \bigg )^{1/2}\nonumber \\\ge & {} \bigg ( \int _{\Delta } d^3x \,|(F^{-1} h)(x)|^2 \bigg )^{1/2}- c|\lambda |^{1/4} \bigg ( \int _{\Delta } d^3x\, (1+\log (1+|x|))^2 \bigg )^{1/2}. \nonumber \\ \end{aligned}$$
(4.75)

For any \(\Delta \) intersecting with the support of \(F^{-1} h\), the first term in the second line of (4.75) is positive and independent of \(\lambda \). As the second term tends to zero as \(\lambda \rightarrow 0\), this concludes the proof. \(\square \)

It remains to prove the following estimates.

Lemma 4.7

Consider the expression

$$\begin{aligned} \gamma _{\textrm{int}}(p,x,t):= 2 \int d^3k\, f_p(k)^2(|k|-k\cdot \nabla E_p) \cos (|k|t-k\cdot x). \end{aligned}$$
(4.76)

The following bounds hold:

  1. (a)

    Fix some \(0<c_0<1\). For any \(M\in {\mathbb {N}}\), there exists a constant \(c_M\), uniform in \(p\in S\), s.t.

    $$\begin{aligned} \sup _{(|x|/t)\le c_0 }| \gamma _{\textrm{int}}(p,x,t) | \le |\lambda |^2\frac{c_M}{t^M}. \end{aligned}$$
    (4.77)
  2. (b)

    For all \(p\in S\) and \((t,x)\in {\mathbb {R}}^4\)

    $$\begin{aligned} | \gamma _{\textrm{int}}(p,x,t) |\le |\lambda |^2 \frac{c}{t} |\log \,t|. \end{aligned}$$
    (4.78)

Analogous estimates hold if we replace \(p\mapsto f_p(k)^2(|k|-k\cdot \nabla E_p)\) in (4.76) by its arbitrary derivatives w.r.t. p.

Proof

Proceeding to spherical coordinates \(d^3k=d\Omega (e_k) |k|^2d|k|\), we have

$$\begin{aligned} \begin{aligned} \gamma _{\textrm{int}}(p,x,t){} & {} = \int d\Omega (e_k) \int _0^{\infty } d|k| \, f(|k|, e_k,p) \cos (|k|t(1- e_k\cdot \textrm{v} )), \\{} & {} f(|k|,e_{k},p):=|\lambda |^2\frac{\chi _{\kappa }(k)^2}{ 2}\frac{1}{(1-e_k\cdot \nabla E_{p} )}, \end{aligned} \end{aligned}$$
(4.79)

where we set \(\textrm{v}:=x/t\). We suppose that \(|\textrm{v}|\le c_0<1\) and consider part (a) of the lemma. By integrating by parts w.r.t. |k| and exploiting that sine vanishes at zero, we obtain

$$\begin{aligned} \gamma _{\textrm{int}}(p,x,t)= -\int d\Omega (e_k) \int _0^{\infty } d|k| \, \partial _{|k|} f(|k|, e_k,p) \frac{1}{t(1- e_k\cdot \textrm{v}) } \sin (|k|t(1- e_k\cdot \textrm{v} )).\nonumber \\ \end{aligned}$$
(4.80)

Now, we can continue integrating by parts, exploiting that \(\partial _{|k|}f\) vanishes in a fixed neighbourhood of zero due to our assumptions on \(\chi _{\kappa }\) (1.3). The fact that \((1- e_k\cdot \textrm{v})\) is never zero in this case gives the claim.

Proceeding to (b), we suppose that \(|\textrm{v}|\ge c_0>0\) as the case \(|\textrm{v}|\le c_0\) is settled by (a). We choose the third axis in the direction of \(\textrm{v}\) and write

$$\begin{aligned}{} & {} \gamma _{\textrm{int}}(p,x,t)\nonumber \\{} & {} \qquad = \int _{|k|\ge 1/t} d|k| \int _0^{2\pi } { d\varphi } \int _{-1}^{1} d\cos (\theta ) \, f(|k|, e(\cos (\theta ), { \varphi }),p) \nonumber \\{} & {} \qquad \quad \times \cos (|k|t(1- |\textrm{v}| \cos (\theta ) ))+O(t^{-1})\nonumber \\{} & {} \qquad = -\int _{|k|\ge 1/t} d|k| \int _0^{2\pi } { d\varphi } \int _{-1}^{1} d\cos (\theta ) \, f(|k|, e(\cos (\theta ), { \varphi }),p) \nonumber \\{} & {} \qquad \quad \times \frac{1}{t|k||v|}\frac{d}{d\cos (\theta )} \sin (|k|t(1- |\textrm{v}| \cos (\theta ) )) +O(t^{-1})\nonumber \\{} & {} \qquad = -\int _{|k|\ge 1/t} d|k| \int _0^{2\pi } { d\varphi } \, f(|k|, e(\cos (\theta ), { \varphi }),p) \nonumber \\{} & {} \qquad \quad \times \frac{1}{t|k||v|}\sin (|k|t(1-|\textrm{v}| \cos (\theta ) ))|^{\cos \theta =1}_{\cos \theta =-1} +O(t^{-1})\nonumber \\{} & {} \qquad \quad +\int _{|k|\ge 1/t} d|k| \int _0^{2\pi } { d\varphi } \int _{-1}^{1} d\cos (\theta ) \,\bigg ( \frac{d}{d\cos (\theta )} f(|k|, e(\cos (\theta ), { \varphi }),p)\bigg )\nonumber \\{} & {} \qquad \times \frac{1}{t|k||v|}\sin (|k|t(1- |\textrm{v}| \cos (\theta ) )). \end{aligned}$$
(4.81)

By estimating \(|\sin (|k|t(1- |\textrm{v}| \cos (\theta ) )) |\le 1\) everywhere above and using that the integration in |k| is over a compact set, the claim follows from

$$\begin{aligned} \frac{1}{t}\int _{\kappa \ge |k|\ge 1/t} d|k| \frac{1}{|k|}\le \frac{c'}{t}|\log (t)|. \end{aligned}$$
(4.82)

This concludes the proof. \(\square \)

5 Conclusions

In this paper, we proposed a new construction of infraparticle states in the massless Nelson model. The approximating sequence does not involve infrared cut-offs and the proof of convergence is relatively simple: Taking the spectral results from [1, 15, 16] for granted, it amounts to the Cook method combined with the stationary phase method, like for basic Schrödinger operators. It is legitimate to ask how the new infraparticle state compares with the established knowledge on the infrared problem in the Nelson model. To partially answer this question, we provide some heuristic remarks on the relation of our states to the Faddeev–Kulish approach. First, we note that the asymptotically dominant part of the wave packet (1.8) should propagate along the ballistic trajectory \(x=\nabla E_p t\), thus \(\psi _t\) should have the same limit as

$$\begin{aligned} \psi ^{\textrm{D}}_t(x):= & {} e^{iHt} \int d^3p\, h(p)\, e^{-i(E_p +H_{\textrm{f}} )t } e^{i \gamma (p,\nabla E_pt,t)} \nonumber \\{} & {} \quad \times W\big ( f_p (1 - e^{i|k|t-ik\cdot \nabla E_p t}) \big ) e^{iH_{\textrm{f}}t } \frac{1}{(2\pi )^{3/2}}e^{i(p- P_{\textrm{f}})\cdot x} \phi _p. \end{aligned}$$
(5.1)

To proceed, let us second quantize also the electrons, denote their creation and annihilation operators by \(b^{(*)}\) and the common vacuum of the electrons and photons by \(\Omega \). Expressing \(\phi _p\in {\mathcal {F}}\) by its n-particle wave functions \({ \phi ^{n}_{p}}\), we define its renormalized creation operator in a standard manner [2]:

$$\begin{aligned} \hat{b}^*_{\textrm{w}}(p):=\sum _{n=0}^{\infty }\frac{1}{ \sqrt{n!} }\int d^{3n}k\, { \phi ^{n}_{p}}(k_1,\ldots , k_{n})a^*(k_1)\ldots a^*(k_{n}) \, b^*(p- (k_1+\cdots +k_{n})),\nonumber \\ \end{aligned}$$
(5.2)

so that \( \frac{1}{(2\pi )^{3/2}} e^{i(p- P_{\textrm{f}})\cdot x} \phi _p\) can be identified with \({\hat{b}}^*_{\textrm{w}}(p)\Omega \). Now recalling that \(f_p(k)=v(k)\frac{1}{|k|-k\cdot \nabla E_p}\), we can write

$$\begin{aligned}{} & {} W\big (f_p(1 - e^{i|k| t-ik\cdot \nabla E_p t}) \big )\nonumber \\{} & {} \quad = \exp {\bigg (-i \int _0^td \tau \, e^{iH_{\textrm{f}}\tau }\big \{ a^*\big ( v e^{ -ik\cdot \nabla E_p \tau } \big ) + a\big ( v e^{ -ik\cdot \nabla E_p \tau } \big ) \big \} e^{-iH_{\textrm{f}}\tau } \bigg ) } \nonumber \\{} & {} \quad =e^{iC_pt} e^{-i\gamma (p,\nabla E_pt,t)} \textrm{Texp}\bigg ({-i \int _0^td \tau \, e^{iH_{\textrm{f}}\tau }\big \{ a^*\big ( v e^{ -ik\cdot \nabla E_p \tau } \big ) + a\big ( v e^{ -ik\cdot \nabla E_p \tau } \big ) \big \} e^{-iH_{\textrm{f}}\tau } \bigg )},\nonumber \\ \end{aligned}$$
(5.3)

where \(C_p:=\int d^3k\,\frac{v(k)^2}{|k|-k\cdot \nabla E_p}\) is finite and the time-ordered exponential \(U_{\textrm{D}}(t)\!:=\textrm{Texp}(\,\ldots \,)\) is the Dollard modifier of the Nelson model, cf [14, formula (3.6)]. Thus, (5.1) can be rewritten as

$$\begin{aligned} \psi ^{\textrm{D}}_t=e^{iHt}\int d^3p\, h(p)\, e^{-i(E_p +H_{\textrm{f}}-C_p)t } U_{\textrm{D}}(t) e^{iH_{\textrm{f}}t }\hat{b}_{\textrm{w}}^*(p)\Omega . \end{aligned}$$
(5.4)

We recall from [14], that a direct application of the Faddeev–Kulish prescription to the Nelson model leads to a formula which differs from (5.4) only by a substitution \(\hat{b}_{\textrm{w}}^*(p)\rightarrow b^*(p)\). We believe that this discrepancy can be attributed to the quantum mechanical origin of the Dollard formalism which makes it difficult to reconcile with the electron mass renormalization present in the model. We think that formula (5.4) is a correct implementation of the Faddeev–Kulish formalism in the Nelson model and hope that the findings of the present paper will lead to a rigorous proof of convergence of \(\psi ^{\textrm{D}}_t\) as \(t\rightarrow \infty \).

There are several other future research directions, which we would like to point out. They include a proof of expected properties of infraparticle states familiar from [33] such as the convergence of asymptotic electron velocity and asymptotic photon fields on our states and the clustering relation, i.e. \(\langle \psi _1^+, \psi ^+_2\rangle =\langle h_1, h_2\rangle \), where the former scalar product is in \({\mathcal {H}}\), the latter in \(L^2({\mathbb {R}}^3_k)\) and \(h_j\) are related to \(\psi _j^+:=\lim _{t\rightarrow \infty }\psi _{j,t}\), \(j=1,2\), via (1.8). These problems appear to be within reach of available methods and are not treated here mainly to keep this paper within reasonable limits. A more intriguing, but still quite tractable problem, is to provide a non-perturbative proof of the Weinberg’s soft photon theorem [38] in the massless Nelson model. Such a proof would provide a useful benchmark to test various perturbative versions, e.g. [23, 31], currently considered in the context of the Strominger’s ‘infrared triangle’ [37]. Another meaningful direction is to apply our construction of infraparticle scattering states to more sophisticated models, such as the Nelson model without the UV cut-off or the Pauli–Fierz model. This direction faces, however, a technical challenge of generalizing the spectral results from [1, 15, 16] to these theories. There is a solid basis for such endeavours, e.g. [4, 5, 8, 9, 20, 24, 25, 28, 29], but also a lot of work remains to be done.