Infraparticle Scattering States in Non- Relativistic QED: I

Communications in Mathematical Physics (2011) DOI 10.1007/s00220-009-0950-x

Thomas Chen, Jurg¨ Frohlich,¨ Alessandro Pizzo Infraparticle Scattering States in Non- Relativistic QED: I. The Bloch-Nordsieck Paradigm

Received: 30 August 2008 / Accepted: 25 August 2009a∗ c TheAuthor(s) 2009

Abstract We construct infraparticle scattering states for Compton scattering in the standard model of non-relativistic QED. In our construction, an infrared cutoff initially introduced to regularize the model is removed completely. We rigorously establish the properties of infraparticle scattering theory predicted in the classic work of Bloch and Nordsieck from the 1930’s, Faddeev and Kulish, and others. Our results represent a basic step towards solving the infrared problem in (non- relativistic) QED.

I Introduction

The construction of scattering states in Quantum Electrodynamics (QED) is an old open problem. The main difficulties in solving this problem are linked to the infa- mous infrared catastrophe in QED: It became clear very early in the development of QED that, at the level of perturbation theory (e.g., for Compton scattering), the transition amplitudes between formal scattering states with charges and a finite number of photons are ill-defined, because, typically, Feynman amplitudes containing vertex or electron self-energy corrections exhibit logarithmic infrared divergences; ??. A pragmatic approach proposed by Jauch and Rohrlich, ??, and by Yennie, Frautschi, and Suura, ?, is to circumvent this difficulty by considering inclusive cross sections: One sums over all possible final states that include photons whose total energy lies below an arbitrary threshold energy ε > 0. Then the infrared divergences due to soft virtual photons are formally canceled by those corresponding to the emission of soft photons of total energy below ε, order by order in perturbation theory in powers of the finestructure constant α. A drawback of this approach becomes apparent when one tries to formulate a scattering theory that

University of Texas at Austin, Austin, TX 78712, USA. [email protected], University of California Davis, Davis, CA 95616, USA. [email protected] Bures sur Yvette, France. [email protected] 2 T. Chen, J. Frohlich,¨ A. Pizzo is ε-independent: Because the transition probability Pε for an inclusive process is estimated to be O(εconst.α ), the threshold energy ε cannot be allowed to approach zero, unless “Bremsstrahlungs” processes (emission of photons) are properly in- corporated in the calculation. An alternative approach to solving the infrared problem is to go beyond inclusive cross sections and to define α-dependent scattering states containing an infinite number of photons (so-called soft-photon clouds), which are expected to yield finite transition amplitudes, order by order in perturbation theory. The works of Chung ?, Kibble ?, and Faddeev and Kulish ?, between 1965 and 1970, represent promising, albeit incomplete progress in this direction. Their approaches are guided by an ansatz identified in the analysis of certain solvable models introduced in early work by Bloch and Nordsieck, ?, and extended by Pauli and Fierz, ?, in the late 1930’s. In a seminal paper ? by Bloch and Nordsieck, it was shown (under certain approximations that render their model solvable) that, in the presence of asymptotic charged particles, the scattering representations of the asymptotic photon field are a coherent non-Fock representation, and that formal scattering states with a finite number of soft photons do not belong to the physical Hilbert space of a system of asymptotically freely moving electrons interacting with the quantized radiation field. These authors also showed that the coherent states describing the soft-photon cloud are parameterized by the asymptotic velocities of the electrons. The perturbative recipes for the construction of scattering states did not re- move some of the major conceptual problems. New puzzles appeared, some of which are related to the problem that Lorentz boosts cannot be unitarily imple- mented on charged scattering states; see ?. This host of problems was addressed in a fundamental analysis of the structural properties of QED, and of the infrared problem in the framework of general quantum field theory; see ?. Subsequent de- velopments in axiomatic quantum field theory have led to results that are of great importance for the topics treated in the present paper: i) Absence of dressed one-electron states with a sharp mass; see ??. ii) Corrections to the asymptotic dynamics, as compared to the one in a theory with a positive mass gap; see ?. iii) Superselection rules pertaining to the space-like asymptotics of the quantized electromagnetic field, and connections to Gauss’ law; see ?. In the early 1970’s, significant advances on the infrared problem were made for Nelson’s model, which describes non-relativistic matter linearly coupled to a scalar field of massless bosons. In ??, the disappearance of a sharp mass shell for the charged particles was established for Nelson’s model, in the limit where an infrared cut-off is removed. (An infrared cutoff is introduced, initially, with the purpose to eliminate the interactions between charged particles and soft boson modes). Techniques developed in ?? have become standard tools in more recent work on non-relativistic QED, and attempts made in ?? have stimulated a deeper understanding of the asymptotic dynamics of charged particles and photons. The analysis of spectral and dynamical aspects of non-relativistic QED and of Nel- son’s model constitutes an active branch of contemporary mathematical physics. In questions relating to the infrared problem, mathematical control of the removal of the infrared cutoff is a critical issue still unsolved in many situations. The construction of an infraparticle scattering theory for Nelson’s model, after removal of the infrared cutoff, has recently been achieved in ? by introducing Infraparticle States in QED I 3 a suitable scattering scheme. This analysis involves spectral results substantially improving those in ?. It is based on a new multiscale technique developed in ?. While the interaction in Nelson’s model is linear in the creation- and annihilation operators of the boson field, it is non-linear and of vector type in non- relativistic QED. For this reason, the methods developed in ?? do not directly apply to the latter. The main goal of the present work is to construct an infraparticle scattering theory for non-relativistic QED inspired by the methods of ??. In a companion paper, ?, we derive those spectral properties of QED that are crucial for our analysis of scattering theory and determine the mass shell structure in the infrared limit. We will follow ideas developed in ?. Bogoliubov transformations, proven in ? to characterize the soft photon clouds in non-relativistic QED, represent an important element in our construction. The proof in ? uses the uniform bounds on the renormalized electron mass previously established in ?. We present a detailed definition of the model of non-relativistic QED in Sect. ??. Aspects of infraparticle scattering theory, developed in this paper, are described in Sect. ??. To understand why free radiation parametrized by the asymptotic velocities of the charged particles must be expected to be present in all the scattering states, we recall a useful point of view based on classical electrodynamics that was brought to our attention by Morchio and Strocchi. We consider a single, classical charged point-particle, e.g., an electron, moving along a world line (t,Ex(t)) in Minkowski space, with Ex(0) =E0. We suppose that, for t ≤ 0, it moves at a constant velocity Evin, and, for t > t¯ > 0, at a constant velocity Evout 6= Evin, |Evout |,|Evin| < c, where c is the speed of light that we set equal to 1. Thus,

Ex(t) = Evin ·t, for t ≤ 0, (I.1) and

Ex(t) = Ex∗ +Evout ·t, for t ≥ t¯, (I.2) for some Ex∗. For times t ∈ [0,t¯], the particle performs an accelerated motion. We propose to analyze the behavior of the electromagnetic field in the vicinity of the particle and the properties of the free electromagnetic radiation at very early times (t → −∞,“in”) and very late times (t → +∞, “out”). For this purpose, we must solve Maxwell’s equations for the electromagnetic field tensor, F µν (t,Ey), given the 4- current density corresponding to the trajectory of the particle; (back reaction of the electromagnetic field on the motion of the charged particle is neglected):

µν ν ∂µ F (t,Ey) = J (t,Ey) (I.3) with

Jν (t,Ey) := −q( δ (3)(Ey −Ex(t)), Ex˙(t)δ (3)(Ey −Ex(t))), (I.4) where, in the units used in our paper, q = 2(2π)3α1/2 (α is the ﬁnestructure constant). We solve Eq. (??) with prescribed spatial asymptotics (|Ey| → ∞): Let 4 T. Chen, J. Frohlich,¨ A. Pizzo

µν F (t,Ey) be a solution of (??) that, to leading order in |Ey|−1 (|Ey| → ∞), ap- [EvL.W.] proaches the Lienard-Wiechert´ ﬁeld tensor for a point-particle with charge −q and a constant velocity EvL.W. at all times. Let us denote the Lienard-Wiechert´ ﬁeld tensor of a point-particle with charge −q moving along a trajectory (t,Ex(t)) in Minkowski space with Ex(0) =: Ex and µν µν Ex˙(t) ≡Ev, for all t, by F (t,Ey). Apparently, we are looking for solutions, F (t,Ey), Ex,Ev [EvL.W.] of (??) with the property that, for all times t,

µν µν |F (t,Ey) − F (t,Ey)| = o(|Ey|−2), (I.5) [EvL.W.] Ex,EvL.W. as |Ey| → ∞, for any Ex. This class of solutions of (??) is denoted by CEvL.W. . It is important to observe that, by causality, the class CEvL.W. is non-empty, for any EvL.W., with |EvL.W.| < 1(= c). This can be seen by choosing Cauchy data for the solution of (??) satisfying (??) at some time t0, e.g., t0 = 0. µν Let us now consider a speciﬁc solution, F (t,Ey), of Eq. (??) in the class [EvL.W.] CEvL.W. . We are interested in the behavior of this solution at very early times (t 0). We expect that, for |Ey −Ex(t)| = o(|t|),

µν µν F (t,Ey) ' F (t,Ey) (I.6) [EvL.W.] E0,Evin (here the symbol ' means: up to a solution of the homogeneous Maxwell equation 1 |y − x(t)| → decaying at least like t2 ). However, for E E ∞,

µν µν F (t,Ey) ' F (t,Ey), (I.7) [EvL.W.] E0,EvL.W. as quantiﬁed in (??). µν We note that, by (??), F (t,Ey) solves Eq. (??), for times t < 0. Thus, E0,Evin

µν µν µν φin (t,Ey) := F (t,Ey) − F (t,Ey) t < 0 (I.8) [EvL.W.] E0,Evin solves the homogenous Maxwell equation, i.e., Eq. (??) with Jν ≡ 0. For t t¯, we expect that, for |Ey −Ex(t)| = o(t),

µν µν F (t,Ey) ' F (t,Ey) (I.9) [EvL.W.] Ex∗,Evout (here the symbol ' means: up to a solution of the homogeneous Maxwell equation 1 |y − x(t)| → decaying at least like t2 ). But, for E E ∞,

µν µν F (t,Ey) ' F (t,Ey), (I.10) [EvL.W.] E0,EvL.W. as quantiﬁed in (??). We note that, by (??), F µν (t,y) solves Eq. (??), for times Ex∗,Evout E t > t¯. Thus,

µν µν µν φ (t,Ey) := F (t,Ey) − F (t,Ey) t > t¯ (I.11) out [EvL.W.] Ex∗,Evout solves the homogenous Maxwell equation. Infraparticle States in QED I 5

µν Next, we recall that φas (t,Ey), with as = in/out, can be derived from an elec- µ tromagnetic vector potential, Aas, by

µν µ ν ν µ φas (t,Ey) = ∂ Aas(t,Ey) − ∂ Aas(t,Ey). (I.12)

µ µ We can impose the Coulomb gauge condition on Aas: Aas = (0,AEas(t,Ey)), with E∇ · AEas(t,Ey) ≡ 0. It turns out (and this can be derived from formulae one ﬁnds, −1 e.g., in ?), that, to leading order in |Ey| (|Ey| → ∞), AEas(t,Ey) is given by

( ∗ ) Z 3 Evas ·Eε 1 d k Ek,λ −iEk·Ey+i|Ek|t AEas(t,Ey) := α 2 Eε e + c.c. ∑ q 3 ˆ Ek,λ λ |Ek| |Ek| 2 (1 −Evas · k) ( ∗ ) Z 3 EvL.W. ·Eε 1 d k Ek,λ −iEk·y+i|Ek|t −α 2 Eε e E + c.c. , (I.13) ∑ q 3 ˆ Ek,λ λ |Ek| |Ek| 2 (1 −EvL.W. · k)

ˆ E where k is the unit vector in the direction of k, and EεEk,+, EεEk,− are transverse polar- ∗ ization vectors with kˆ ·Eε = 0, λ = +,−, and Eε ·Eε = δ 0 . Ek,λ Ek,λ Ek,λ 0 λ,λ The free ﬁeld

µν µν µν φ (t,Ey) = φout (t,Ey) − φin (t,Ey) (I.14) is the radiation emitted by the particle due to its accelerated motion, as t → ∞. It is well known that Eqs. (??)-( ??) can be made precise within classical electrodynamics under some standard assumptions on the Cauchy data for the solutions in addition to the condition in Eq. (??). We will see that analogous statements also hold in our model of quantum electrodynamics with non-relativistic matter. In this paper, we treat the quantum theory of a system consisting of a nonrelativistic charged particle only interacting with the quantized e.m. field. The motion of the quantum particle depends on the back-reaction of the field, and the asymptotic in- and out-velocities of this particle are not attained at finite times. However, the infrared features of the asymptotic radiation in the classical model, described above for a given current, are reproduced in this interacting quantum model.

In fact, the set of classes CEvL.W. , associated with different currents but at fixed EvL.W., corresponds to one of the superselection sectors of the quantized theory; see e.g. [3]. In particular, the Fock representation, which is the usual (but not the only possible) choice for the representation of the algebra of photon creation- and annihilation operators, corresponds to EvL.W. = 0. This implies that, in the Fock representation of the interpolating photon creation- and annihilation operators, an infrared-singular asymptotic electromagnetic-field configuration must be present for all values of the asymptotic velocity of the electron different from zero. In particular, after replacing the classical velocities with the spectral values of the quantum operators Evout/in, the background field with EvL.W. = 0 (given by (??)) corresponds to the background radiation described by the coherent non-Fock representations of the algebra of asymptotic photon creation- and annihilation operators labeled by Evout/in; see also Sect. ??. 6 T. Chen, J. Frohlich,¨ A. Pizzo

II Deﬁnition of the Model

The Hilbert space of pure state vectors of the system consisting of one non- relativistic electron interacting with the quantized electromagnetic ﬁeld is given by

H := Hel ⊗ F , (II.1) 2 3 where Hel = L (R ) is the Hilbert space for a single electron; (for expository con- venience, we neglect the spin of the electron). The Fock space used to describe the states of the transverse modes of the quantized electromagnetic field (the photons) in the Coulomb gauge is given by ∞ M (N) (0) F := F , F = CΩ , (II.2) N=0 where Ω is the vacuum vector (the state of the electromagnetic field without any excited modes), and N (N) O F := SN h , N ≥ 1, (II.3) j=1 where the Hilbert space h of a single photon is 2 3 h := L (R × Z2). (II.4) 3 Here, R is momentum space, and Z2 accounts for the two independent transverse polarizations (or helicities) of a photon. In (??), SN denotes the orthogonal NN projection onto the subspace of j=1 h of totally symmetric N-photon wave functions, to account for the fact that photons satisfy Bose-Einstein statistics. Thus, F (N) is the subspace of F of state vectors for configurations of exactly N photons. In this paper, we use units such that Planck’s constant h¯, the speed of light c, and the mass of the electron are equal to unity. The dynamics of the system is generated by the Hamiltonian 2 E 1/2 −i∇Ex + α AE(Ex) H := + H f . (II.5) 2 The multiplication operator Ex ∈ R3 corresponds to the position of the elec- E ∼ tron. The electron momentum operator is given by Ep = −i∇Ex; α = 1/137 is the finestructure constant (which, in this paper, is treated as a small parameter), AE(Ex) denotes the (ultraviolet regularized) vector potential of the transverse modes of the quantized electromagnetic field at the point Ex (the electron position) in the Coulomb gauge, E ∇Ex · AE(Ex) = 0. (II.6) H f is the Hamiltonian of the quantized, free electromagnetic field, given by Z H f := d3k |Ek|a∗ a , (II.7) ∑ Ek,λ Ek,λ λ=± Infraparticle States in QED I 7 where a∗ and a are the usual photon creation- and annihilation operators, Ek,λ Ek,λ which satisfy the canonical commutation relations

∗ 0 [a , a ] = δ 0 δ(Ek −Ek ) , (II.8) Ek,λ Ek0,λ 0 λλ [a# , a# ] = 0, (II.9) Ek,λ Ek0,λ 0 with a# = a or a∗. The vacuum vector Ω obeys the condition

aEk,λ Ω = 0 , (II.10)

3 for allEk ∈ R and λ ∈ Z2 ≡ {+,−}. The quantized electromagnetic vector potential is given by

Z d3k n o AE(y) := Eε e−iEk·Eya∗ + Eε ∗ eiEk·Eya , (II.11) E ∑ q Ek,λ Ek,λ Ek,λ Ek,λ λ=± BΛ |Ek|

3 where EεEk,−, EεEk,+ are photon polarization vectors, i.e., two unit vectors in R ⊗C satisfying

Eε ∗ ·Eε = δ , Ek ·Eε = 0 , (II.12) Ek,λ Ek,µ λ µ Ek,λ

E for λ, µ = ±. The equation k ·EεEk,λ = 0 expresses the Coulomb gauge condition. Moreover, BΛ is a ball of radius Λ centered at the origin in momentum space; Λ represents an ultraviolet cutoff that will be kept fixed throughout our analysis. The vector potential defined in (??) is thus regularized in the ultraviolet. Throughout this paper, it will be assumed that Λ ≈ 1 (the rest energy of an electron), and that α is sufficiently small. Under these assumptions, the Hamito- nian H is selfadjoint on D(H0), the domain of definition of the operator

(−iE∇ )2 H := Ex + H f . (II.13) 0 2

The perturbation H − H0 is small in the sense of Kato. The operator measuring the total momentum of a state of the system consisting of the electron and the electromagnetic ﬁeld is given by

PE := Ep + PE f , (II.14) E where Ep = −i∇Ex is the momentum operator for the electron, and Z PE f := d3kEka∗ a (II.15) ∑ Ek,λ Ek,λ λ=± is the momentum operator for the radiation ﬁeld. The operators H and PE are essentially selfadjoint on the domain D(H0), and since the dynamics is invariant under translations, they commute: [H,PE] = 0. The 8 T. Chen, J. Frohlich,¨ A. Pizzo

Hilbert space H can be decomposed on the joint spectrum, R3, of the component- operators of PE. Their spectral measure is absolutely continuous with respect to Lebesgue measure,

Z ⊕ 3 H := HPE d P, (II.16) where each ﬁber space HPE is a copy of Fock space F .

Remark Throughout this paper, the symbol PE stands for both a variable in R3 and a vector operator in H , depending on the context. Similarly, a double meaning is also associated with functions of the total momentum operator. (E.g.: In Eq. (??) Eσ is an operator on the Hilbert space , while in Eq. (??) it is a function of PE H PE ∈ R3.)

To each ﬁber space HPE there corresponds an isomorphism

b IPE : HPE −→ F , (II.17) where F b is the Fock space corresponding to the annihilation- and creation operators b , b∗ , where b is given by eiEk·Exa , and b∗ by e−iEk·Exa∗ , with vac- Ek,λ Ek,λ Ek,λ Ek,λ Ek,λ Ek,λ iPE·Ex uum Ω f = IPE(e ), where Ex is the electron position. To deﬁne IPE more precisely, we consider an (improper) vector ψ( f (n);PE) ∈ HPE with a deﬁnite total momentum, which describes an electron and n photons. Its wave function, in the variables (Ex;Ek1,...,Ekn;λ1,...,λn), is given by

i(PE−Ek1−···−Ekn)·Ex (n) e f (Ek1,λ1;······ ;Ekn,λn), (II.18)

(n) where f is totally symmetric in its n arguments. The isomorphism IPE acts by way of

E E i(PE−k1−···−kn)·Ex (n) E E IPE e f (k1,λ1;······ ;kn,λn) (II.19) 1 Z 3 3 (n) E E ∗ ∗ = √ ∑ d k1 ...d kn f (k1,λ1;······ ;kn,λn)bE ···bE Ω f . n! k1,λ1 kn,λn λ1,...,λn (II.20)

The Hamiltonian H maps each ﬁber space HPE into itself, i.e., it can be written as Z ⊗ 3 H = HPE d P, (II.21) where

HPE : HPE −→ HPE. (II.22) Infraparticle States in QED I 9

Written in terms of the operators b , b∗ , and of the variable PE, the fiber Hamil- Ek,λ Ek,λ tonian HPE has the form 2 PE − PE f + α1/2AE H := + H f , (II.23) PE 2 where Z PE f = d3kEkb∗ b , (II.24) ∑ Ek,λ Ek,λ λ Z H f = d3k|Ek|b∗ b , (II.25) ∑ Ek,λ Ek,λ λ and Z d3k n o AE := b∗ Eε + Eε ∗ b . (II.26) ∑ q Ek,λ Ek,λ Ek,λ Ek,λ λ BΛ |Ek| Let 1 S := {PE ∈ 3 : |PE| < }. (II.27) R 3 In order to give precise meaning to the constructions used in this work, we restrict the total momentum PE to the set S , and we introduce an infrared cut-off at an energy σ > 0 in the vector potential. The removal of the infrared cutoff in the construction of scattering states is the main problem solved in this paper. The restriction of PE to S guarantees that the propagation speed of a dressed electron is strictly smaller than the speed of light. However, our results can be extended to 1 a region S (inside the unit ball) of radius larger than 3 . We start by studying a regularized fiber Hamiltonian given by 2 PE − PE f + α1/2AEσ Hσ := + H f (II.28) PE 2 E acting on the fiber space HPE, for P ∈ S , where Z d3k n o AEσ := b∗ Eε + Eε ∗ b , (II.29) ∑ q Ek,λ Ek,λ Ek,λ Ek,λ λ BΛ \Bσ |Ek| and where Bσ is a ball of radius σ. Remark In a companion paper ?, we construct dressed one-electron states of fixed σ σ momentum given by the ground state vectorsΨ j of the Hamiltonians H j , and we PE PE σ σ 0 compare ground state vectors Ψ j , Ψ j corresponding to different fiber Hamilto- PE PE0 σ σ 0 nians H j , H j with PE 6= PE0. We compare these ground state vectors as vectors in PE PE0 the Fock space F b. In the sequel, we use the expression σ σ 0 kΨ j −Ψ j k (II.30) PE PE0 F 10 T. Chen, J. Frohlich,¨ A. Pizzo as an abbreviation for

σ j σ j0 kI (Ψ ) − I 0 (Ψ )k ; (II.31) PE PE PE PE0 F k · k stands for the Fock norm. Holder¨ continuity properties of Ψ σ in σ and in F PE PE are proven in ?. These properties play a crucial role in the present paper.

II.1 Summary of contents

In Sect. ??, time-dependent vectors ψh,κ (t) approximating scattering states are constructed, and the main results of this paper are described, along with an outline of infraparticle scattering theory. In Sects. ?? and ??, ψh,κ (t) is shown to converge out/in to a scattering state ψh,κ in the Hilbert space H , as time t tends to inﬁnity. This out/in result is based on mathematical techniques introduced in ?. The vector ψh,κ represents a dressed electron with a wave function h on momentum space whose support is contained in the set S (see details in Sect. ??), accompanied by a cloud of soft photons described by a Bloch-Nordsieck operator, and with an upper cutoff κ imposed on photon frequencies. This cutoff can be chosen arbitrarily. In Sect. ??, we construct the scattering subspaces H out/in. Vectors in these subspaces are obtained from certain subspaces, H˚ out/in, by applying “hard” asymptotic photon creation operators. These spaces carry representations of the algebras out/in out/in Aph and Ael of asymptotic photon creation- and annihilation operators and asymptotic electron observables, respectively, which commute with each other. The latter property proves asymptotic decoupling of the electron and photon dynamics. We rigorously establish the coherent nature and the infrared properties of out/in the representation of Aph identiﬁed by Bloch and Nordsieck in their classic paper, ?. In a companion paper ?, we establish the main spectral ingredients for the construction and convergence of the vectors {Ψ σ }, as σ tends to 0. These results PE are obtained with the help of a new multiscale method introduced in ?, to which we refer the reader for some details of the proofs. In the Appendix, we prove some technical results used in the proofs.

III Infraparticle Scattering States

Infraparticle scattering theory is concerned with the asymptotic dynamics in QFT models of infraparticles and massless fields. Contrary to theories with a non- vanishing mass gap, the picture of asymptotically freely moving particles in the Fock representation is not valid, due to the inseparability of the dynamics of charged massive particles and the soft modes of the massless asymptotic fields. Our starting point is to study the (dressed one-particle) states of a (non-relativistic) electron when the interactions with the soft modes of the photon field are turned off. We then analyze their limiting behavior when this infrared cut-off is removed. Infraparticle States in QED I 11

This amounts to studying vectors ψσ , σ > 0, in the Hilbert space H that are solutions to the equation

Hσ σ = Eσ σ , ψ PE ψ (III.1) where Hσ = R ⊕ Hσ d3P, and Eσ is a function of the vector operator PE; Eσ is PE PE PE the electron energy function defined more precisely in Sect. ??. Since in our model non-relativistic matter is coupled to a relativistic field, the form of Eσ is PE not fixed by symmetry, except for rotation invariance. Furthermore, the solutions of (??) give rise to vectors in the physical Hilbert space describing wave packets of dressed electrons of the form Z σ (h) = h(PE) σ d3P, ψ ΨPE (III.2) where the support of h is contained in a ball centered at PE = 0, chosen such that |E∇Eσ | < 1, as a function of PE, i.e., we must impose the condition that the maxi- PE mal group velocity of the electron which, a priori, is not bounded from above in our non-relativistic model, is bounded by the speed of light. (For group velocities larger than the velocity of light, the one-electron states decay by emission of Cerenkov radiation.) The guiding principle motivating our analysis of limiting or improper one- particle states, ψσ (h) for σ → 0, is that refined control of the infrared singularities, which push these vectors out of the space H , as σ → 0, should enable one to characterize the soft photon cloud encountered in the scattering states. The analysis of Bloch and Nordsieck, ?, suggests that the infrared behavior of the state describing the soft photons accompanying an electron should be singular (i.e., not square- integrable at the origin in photon momentum space), and that it should be determined by the momentum of the asymptotic electron. In mathematical terms, this means that the asymptotic electron velocity is expected to determine an asymptotic Weyl operator (creating a cloud of asymptotic photons), which when applied to a dressed one-electron state ψσ=0(h) yields a well defined vector in the Hilbert space H . This vector is expected to describe an asymptotic electron with wave function h surrounded by a cloud of infinitely many asymptotic free photons, in accordance with the observations sketched in (??)–(??). Our goal in this paper is to translate this physical picture into rigorous mathe- matics, following suggestions made in ? and methods developed in ????.

III.1 Key spectral properties

In our construction of scattering states, we make extensive use of a number of spectral properties of our model proven in ?, and summarized in Theorem ?? below; (they are analogous to those used in the analysis of Nelson’s model in ?). We deﬁne the energy of a dressed one-electron state of momentum PE by

Eσ = Hσ , E = H = Eσ=0. PE infspec PE PE infspec PE PE (III.3) 12 T. Chen, J. Frohlich,¨ A. Pizzo

We refer to Eσ as the ground state energy of the fiber Hamiltonian Hσ . We assume PE PE that the finestructure constant α is so small that |E Eσ | < < ∇ PE νmax 1 (III.4) E E 3 E 1 for all P ∈ S := {P ∈ R : |P| < 3 }, for some constant νmax < 1, uniformly in σ. Corresponding to E∇Eσ , we introduce a Weyl operator PE   Z E∇Eσ 1 3 E ∗ W (E∇Eσ ) := exp α 2 d k P · (Eε b − h.c.) , (III.5) σ PE  ∑ 3 Ek,λ Ek,λ  B \B E 2 λ Λ σ |k| δPE,σ (bk) where Ek E σ δE (bk) := 1 − ∇E · , (III.6) P,σ PE |Ek| acting on HPE, which is unitary for σ > 0. We consider the transformed fiber Hamiltonian Kσ = W (E Eσ )HσW ∗(E Eσ ). PE : σ ∇ PE PE σ ∇ PE (III.7) We note that conjugation by W (E∇Eσ ) acts on the creation- and annihilation op- σ PE erators as a linear Bogoliubov transformation (translation)

1 (Ek) W (E∇Eσ )b# W ∗(E∇Eσ ) = b# − α1/2 σ,Λ E∇Eσ ·Eε # , (III.8) σ PE Ek,λ σ PE Ek,λ 3 PE Ek,λ E 2 |k| δPE,σ (bk) where 1σ,Λ (Ek) stands for the characteristic function of the set BΛ \Bσ . Our methods rely on proving regularity properties in σ and PE of the ground state vector, Φσ , and of the ground state energy, Eσ , of Kσ . These regularity properties are PE PE PE summarized in the following theorem, which is the main result of the companion paper ?.

Theorem III1 For PE ∈ S and for α > 0 sufﬁciently small, the following statements hold. ( 1) The energy Eσ is a simple eigenvalue of the operator Kσ on b. Let I PE PE F Bσ := 3 2 3 {Ek ∈ R ||Ek| ≤ σ}, and let Fσ denote the Fock space over L ((R \Bσ )× σ 2 Z2). Likewise, we deﬁne F0 to be the Fock space over L (Bσ × Z2); hence b = ⊗ σ . On , the operator Kσ has a spectral gap of F Fσ F0 Fσ PE size ρ−σ or larger, separating Eσ from the rest of its spectrum, for some PE constant ρ−, with 0 < ρ− < 1. The contour ρ−σ γ := {z ∈ ||z − Eσ | = } , σ > 0 (III.9) C PE 2 Infraparticle States in QED I 13

bounds a disc which intersects the spectrum of Kσ | in only one point, PE Fσ {Eσ }. The ground state vectors of the operators Kσ are given by PE PE

1 R 1 2π i γ Kσ −z dzΩ f σ = PE ΦPE : 1 R 1 (III.10) k 2π i γ Kσ −z dzΩ f kF PE

b and converge strongly to a non-zero vector ΦPE ∈ F , in the limit σ → 0. 1 (1− ) The rate of convergence is at least of order σ 2 δ , for any 0 < δ < 1. (Although it is not relevant for the purposes of this paper, we note that the results in ? imply the uniformity in δ of the range of values of α, where 1 (1− ) the rate estimate σ 2 δ holds; analogous conclusions follow for the rate estimates below.) The dependence of the ground state energies Eσ of the ﬁber Hamiltonians PE Kσ on the infrared cutoff σ is characterized by the following estimates: PE

0 |Eσ − Eσ | ≤ ( ), PE PE O σ (III.11) and

0 1 (1− ) |E Eσ −E Eσ | ≤ ( 2 δ ), ∇ PE ∇ PE O σ (III.12)

for any 0 < δ < 1, with σ > σ 0 > 0. (I 2) The following Holder¨ regularity properties in PE ∈ S hold uniformly in σ ≥ 0:

σ σ 1 −δ 0 kΦ − Φ k ≤ C 0 |∆PE| 4 (III.13) PE PE+∆PE F δ

and

σ σ 1 −δ 00 |E∇E −E∇E | ≤ C 00 |∆PE| 4 , (III.14) PE PE+∆PE δ

00 0 1 E E E for any 0 < δ < δ < 4 , with P, P + ∆P ∈ S , where Cδ 0 and Cδ 00 are ﬁnite constants depending on δ 0 and δ 00, respectively. (I 3) Given a positive number νmin, there are numbers rα = νmin + O(α) > 0 E and νmax < 1 such that, for P ∈ S \Brα and for α sufﬁciently small,

> > |E Eσ | > > , 1 νmax ∇ PE νmin 0 (III.15)

uniformly in σ. (We also notice that the control on the second derivative of Eσ in PE uniformly in the sharp infrared cut-off σ ≥ 0 (see ?) would allow PE us to take ν ≡ 0,r ≡ 0, and to include electron velocities E∇Eσ arbi- min α PE trarily close to 0, but we prefer to work with an assumption self-contained in the paper). 14 T. Chen, J. Frohlich,¨ A. Pizzo

(I 4) For PE ∈ S and for anyEk 6= 0, the following inequality holds uniformly in σ, for α small enough:

Eσ > Eσ −C |Ek|, (III.16) PE−Ek PE α

where Eσ := infspecHσ and 1 < C < 1, with C → 1 as α → 0. PE−Ek PE−Ek 3 α α 3 ( 5) Let Ψ σ ∈ denote the ground state vector of the ﬁber Hamiltonian Hσ , I PE F PE so that Ψ σ σ σ PE Φ = ζWσ (E∇E ) ζ ∈ C, |ζ| = 1. (III.17) PE PE kΨ σ k PE F

For PE ∈ S , one has that

Ψ σ 1 (Ek) kb PE k ≤ C α1/2 σ,Λ , (III.18) Ek,λ kΨ σ k F E 3/2 PE F |k| where Ψ σ is the ground state of Hσ and C is a positive constant; see PE PE Lemma 6.1 of ? which can be extended toEk ∈ R3 using (I 4). Detailed proofs of Theorem ?? based on results in ?? are given in ?.

III.2 Deﬁnition of the approximating vector Ψh,κ (t)

We construct infraparticle scattering states by using a time-dependent approach to scattering theory. We deﬁne a time-dependent approximating vector ψh,κ (t) that converges to an asymptotic vector, as t → ∞. It describes an electron with wave function h (whose momentum space support is contained in S ), and a cloud of asymptotic free photons with an upper photon frequency cutoff 0 < κ ≤ Λ. This interpretation will be justiﬁed a posteriori. We closely follow an approach to infraparticle scattering theory developed for Nelson’s model in ?, (see also ?). In the context of the present paper, our task is to give a mathematically rigorous meaning to the formal expression

out iHt −iHσ t σ Φκ (h) := lim lim e Wκ,σ (Ev(t),t)e ψ (h), (III.19) t→∞ σ→0 where

 ∗ −i|Ek|t ∗ i|Ek|t  3 Ev(t) ·{Eε a e −Eε a e } 1 Z d k Ek,λ Ek,λ Ek,λ Ek,λ Wκ,σ (Ev(t),t) := expα 2 . ∑ q E λ Bκ \Bσ |Ek| |k|(1 −bk ·Ev(t))

The operator Ev(t) is not known a priori; but, in the limit t → ∞, it must converge to the asymptotic velocity operator of the electron. The latter is determined by E the operator ∇EPE, applied to the (non-Fock) vectors ΨPE. This can be seen by ﬁrst Infraparticle States in QED I 15 considering the infrared regularized model, with σ > 0, which has dressed one- electron states ψσ (h) in H , and by subsequently passing to the limit σ → 0. Formally, for σ → 0, the Weyl operator

iHt −iHσ t e Wκ,σ (Ev(t),t)e (III.20) is an interpolating operator used in the L.S.Z. (Lehmann-Symanzik-Zimmermann) approach to scattering theory for the electromagnetic field, where the photon test functions (in the operator Wκ,σ (Ev(t),t)) are evolved backwards in time with the free evolution, and the photon creation- and annihilation operators are evolved forward in time with the interacting time evolution. Moreover, the photon test functions in (??) coincide with the test functions in the Weyl operator W (E∇Eσ ) σ PE defined in (??), after replacing the operator E∇Eσ by the operator v(t). We stress PE E that, while the Weyl operator W (E∇Eσ ) leaves the fiber spaces invariant, the σ PE HPE ∗ Weyl operator Wκ,σ (Ev(t),t) is expressed in terms of the operators {a, a }, as it must be when describing real photons in a scattering process, and hence does not preserve the fiber spaces. Guided by the expected relation between v(t) and E∇Eσ , as t → ∞ and σ → 0, E PE two key ideas used to make (??) precise are to render the infrared cut-off time- 3 dependent, with σt → 0, as t → ∞, and to discretize the ball S = {PE ∈ R ||PE| < 1 3 }, with a grid size decreasing in time t. This discretization also applies to the velocity operator Ev(t) in expression (??). The existence of infraparticle scattering states in H is established by proving that the corresponding sequence of time-dependent approximating scattering states, which depend on the cutoff σt and on the discretization, defines a strongly convergent sequence of vectors in H . This is accomplished by appropriately tun- ing the convergence rates of σt and of the discretization of S . Our sequence of approximate infraparticle scattering states is defined as follows (for t 1): i) We consider a wave function h with support in a region R which is a union

of cubes contained in S \Brα ; (see condition (I 3) in Theorem ??). We introduce a time-dependent cell partition G (t) of R. This partition is con- L structed as follows: At time t, the linear dimension of each cell is 2n , where L is the diameter of R, and n ∈ N is such that n 1 n+1 1 (2 ) ε ≤ t < (2 ) ε , (III.21)

for some ε > 0 to be ﬁxed later. Thus, the total number of cells in G (t) is 3n ε (t) N(t) = 2 , where n = blog2 t c;(bxc extracts the integer part of x). By G j , we denote the jth cell of the partition G (t). ii) For each cell, we consider a one-particle state of the Hamiltonian Hσt , Z ψ(t) := h(PE)Ψ σt d3P, (III.22) j,σt (t) PE G j where E 1 • h(P) ∈ C0(S \Brα ), with supph ⊆ R; −β • σt := t , for some exponent β (> 1) to be ﬁxed later; 16 T. Chen, J. Frohlich,¨ A. Pizzo

• in (??), the ground state vector, Ψ σt , of Hσt is deﬁned by PE PE Ψ σt := W ∗ (∇Eσt )Φσt , (III.23) PE σt PE PE where Φσt is the ground state of Kσt ; (see Theorem ??). PE PE (t) iii) With each cell G j we associate a soft-photon cloud described by the following “LSZ (Lehmann-Symanzik-Zimmermann) Weyl operator”:

iHt −iHσt t e Wκ,σt (Ev j,t)e , (III.24) where  ∗ −i|Ek|t ∗ i|Ek|t  3 Ev j ·{Eε a e −Eε a e } 1 Z d k Ek,λ Ek,λ Ek,λ Ek,λ 2 Wκ,σt (Ev j,t) := expα . ∑ q E λ Bκ \Bσt |Ek| |k|(1 −bk ·Ev j) (III.25) • Here κ, with 0 < κ ≤ Λ, is an arbitrary (but ﬁxed) photon energy threshold or counter threshold. E σt ∗ • Ev j ≡ ∇E (PEj ) is the c-number vector corresponding to the value of the (t) E σt ∗ “velocity” ∇E (PE) in the center, PEj , of the cell G j . iv) For each cell, we consider a time-dependent phase factor

σ iγ (Ev ,E∇E t ,t) e σt j PE , (III.26) with Z t Z γ (v ,E∇Eσt ,t) := −α E∇Eσt · ΣE (Ek) cos(Ek ·E∇Eσt τ − |Ek|τ)d3kdτ, σt Ej PE PE Ev j PE 1 B S \Bσt στ (III.27) and l l0 l k k l0 1 Σ (Ek) := 2 (δl,l0 − )v . (III.28) Ev j ∑ 2 j 2 l0 |Ek| |Ek| (1 −bk ·Ev j) S −θ Here, στ := τ , and the exponent 0 < θ < 1 will be chosen later. Note that, in (??), (??), E∇Eσt is interpreted as an operator. PE v) The approximate scattering state at time t is given by the expression

N(t) σt σt iHt iγ (Ev j,E∇E ,t) −iE t (t) (t) := e (v ,t)e σt PE e PE , (III.29) ψh,κ ∑ Wκ,σt Ej ψ j,σt j=1

where N(t) is the number of cells in G (t). σ iγ (Ev ,E∇E t ,t) The role played by the phase factor e σt j PE is similar to that of the Coulomb phase in Coulomb scattering. However, in the present case, the phase has a limit, as t → ∞, and is introduced to control an oscillatory term in the Cook argument which is not absolutely convergent (see Sect. ??). Infraparticle States in QED I 17

L Fig. 1 R can be described as a union of cubes with sides of length 2n0 , for some n0 < ∞

III.3 Statement of the main result

The main result of this paper is Theorem ??, below, from which the asymptotic picture described in Sect. ??, below, emerges. It relies on the assumptions summarized in the following hypothesis. Main Assumption III1 The following assumptions hold throughout this paper: (1) The conserved momentum PE takes values in S ; see (??). (2) The ﬁnestructure constant α satisﬁes α < αc, for some small constant αc 1 independent of the infrared cutoff. (3) The wave function h is supported in a set R and is of class C1, where R is

contained in S \Brα , as indicated in Fig. ??, and rα is introduced in (I 3). Theorem III2 Given the Main Assumption ??, the following holds: There exist positive real numbers β > 1, θ < 1 and ε > 0 such that the limit (out) s − lim ψh,κ (t) =: ψ (III.30) t→+∞ h,κ

(where ψh,κ (t) is deﬁned in Eq. (??) and κ, see (??), is the threshold frequency) (out) 2 R E 2 3 exists as a vector in H , and kψh,κ k = |h(P)| d P. Furthermore, the rate of 0 convergence is at least of order t−ρ , for some ρ0 > 0. We note that this result corresponds to Theorem 3.1 of ? for Nelson’s model. The limiting state is the desired infraparticle scattering state without infrared cut-offs. We shall verify that {ψh,κ (t)} is a Cauchy sequence in H , as t → +∞; (or t → −∞). In Sect. ??, we outline the key mechanisms responsible for the convergence of the approximating vectors ψh,κ (t), as t → ∞. We note that, in (??), three different convergence rates are involved: −β • The rate t related to the fast infrared cut-off σt ; −θ S • the rate t , related to the slow infrared cut-off σt (see (??)); • the rate t−ε of the grid size of the cell partition. We anticipate that, in order to control the interaction, • β has to be larger than 1, due to the time-energy uncertainty principle. • The exponent θ has to be smaller than 1, in order to ensure the cancelation of some “infrared tails” discussed in Sect. ??. • The exponent ε, which controls the rate of reﬁnement of the cell decomposition, will have to be chosen small enough to be able to prove certain decay estimates.

III.4 Strategy of convergence proof

Here we outline the key mechanisms used to prove that the approximating vectors ψh,κ (t) converge to a nonzero vector in H , as t → ±∞. 18 T. Chen, J. Frohlich,¨ A. Pizzo

Among other things, we will prove that Z 2 3 1 lim kψh,κ (t)k = khk2 := ( |h(PE)| d P) 2 . (III.31) t→∞ From its deﬁnition, see (??), one sees that the square of the norm of the vector (t) ψh,κ (t) involves a double sum over cells of the partitions G , i.e.,

N(t) σt σt 2 D iγσ (Ev ,E∇E ,t) −iE t (t) ∗ kψ (t)k = e t l PE e PE ψ , (v ,t) h,κ ∑ l,σt Wκ,σt El l, j=1 σ σ iγ (Ev ,E∇E t ,t) −iE t t (t) E × (v ,t)e σt j PE e PE , (III.32) Wκ,σt Ej ψ j,σt where the individual terms, labeled by (l, j), are inner products between vectors (t) (t) (t) labeled by cells Gl and G j of G . A heuristic argument to see where (??) comes from is as follows. Assuming that

• the vectors ψh,κ (t) converge to an asymptotic vector of the form

iHt −iHσ t σ out/in σ=0 lim lim e Wκ,σ (Ev(t),t)e ψ (h) = W , = (Ev(±∞))ψ (h), t→±∞ σ→0 κ σ 0 (III.33)

where

 out/in∗ ∗ out/in  3 Ev(±∞) ·{Eε a −Eε a } out/in 1 Z d k Ek,λ Ek,λ Ek,λ Ek,λ W (Ev(±∞)) := expα 2 , κ,σ=0 ∑ q E λ Bκ |Ek| |k|(1 −bk ·Ev(±∞))

and aout/in∗, aout/in are the creation- and annihilation operators of the asymp- Ek,λ Ek,λ totic photons; • the operators Ev(±∞) commute with the algebra of asymptotic creation- and annihilation operators {aout/in∗, aout/in}; (this can be expected to be a conse- Ek,λ Ek,λ quence of asymptotic decoupling of the photon dynamics from the dynamics of the electron); • the restriction of the asymptotic velocity operators, Ev(±∞), to the improper E dressed one-electron state is given by the operator ∇EPE, i.e., E Ev(±∞)ΨPE ≡ ∇EPEΨPE ; (III.34) then, the two vectors

σ σ σ σ iγ (Ev ,∇E t ,t) −iE t t (t) iγ (Ev ,∇E t ,t) −iE t t (t) (v ,t)e σt j PE e PE ψ and (v ,t)e σt l PE e PE ψ Wκ,σt Ej j,σt Wκ,σt El l,σt (III.35) corresponding to two different cells of G (t) (i.e., j 6= l) turn out to be orthogonal in the limit t → ±∞. One can then show that the diagonal terms in the sum (??) Infraparticle States in QED I 19 are the only ones that survive in the limit t → ∞. The fact that their sum converges 2 to khk2 is comparatively easy to prove. A mathematically precise formulation of this mechanism is presented in Sect. ??. In Sect. ??, part A., the analysis of the scalar products between the cell vectors in (??) is reduced to the study of an ODE. To prove (??), we invoke the following properties of the one-particle states ψ(t) and ψ(t) located in the jth and lth cell, j,σt l,σt respectively: • Their spectral supports with respect to the momentum operator PE are disjoint up to sets of measure zero. • They are vacua for asymptotic annihilation operators, as long as an infrared λ cut-off σt for a ﬁxed time t is imposed: For Schwartz test functions g , we deﬁne Z out/in iHσt s λ i|Ek|s −iHσt s 3 aσ (g) := lim e aE g (Ek)e e d k, (III.36) t s→±∞ ∑ k,λ λ

on the domain of Hσt . An important step in the proof of (??) is to control the decay in time of the off-diagonal terms. After completion of this step, one can choose the rate, t−ε , by which the diameter of the cells of the partition G (t) tends to 0 in such a way that the sum of the off-diagonal terms vanishes, as t → ∞. Precise control is achieved in Sect. ??, part B., where we invoke Cook’s argument and analyze the decay in time s of

d σ E σt σt iH t s iγσt (Ev j,∇E ,s) −iE s (t) e W , (Ev j,s)e PE e PE ψ (III.37) ds κ σt j,σt σt σt iHσt s σ iγ (Ev j,E∇E ,s) −iE s (t) = ie [H t , (v ,s)]e σt PE e PE (III.38) I Wκ,σt Ej ψ j,σt E σt σ dγσt (Ev j,∇E ,s) E σt σt iH t s PE iγσt (Ev j,∇E ,s) −iE s (t) +ie W , (Ev j,s) e PE e PE ψ , (III.39) κ σt ds j,σt for a ﬁxed infrared cut-off σt , and a ﬁxed partition. As we will show, the term in (??) can be written (up to a unitary operator) as

1 Z Z σt 3 3 iγσ (Ev j,E∇E ,s) (t) 2 d y JE (s,y) · E (q)cos(q · y − |q|s)d q e t PE α σt E ΣEv j E E E E ψ j,σt Bκ \Bσt (III.40) plus subleading terms, where JEσt (s,Ey) is essentially the electron current at time s, which is proportional to the velocity operator

1 t i[Hσt , Ex] = Ep + α 2 AEσ (Ex). (III.41)

In (??), the electron current is smeared out with the vector function Z E 3 Egt (s,Ey) := ΣEv j (Eq) cos(Eq ·Ey − |Eq|s)d q, (III.42) Bκ \Bσt 20 T. Chen, J. Frohlich,¨ A. Pizzo which solves the wave equation

s,EyEgt (s,Ey) = 0, (III.43)

σ iγ (Ev ,E∇E t ,s) (t) and is then applied to the one-particle state e σt j PE . Because of the dis- ψ j,σt persive properties of the dynamics of the system, the resulting vector is expected to converge to 0 in norm at an integrable rate, as s → ∞. An intuitive explanation proceeds as follows:

i) A vector function Egt (s,Ey) that solves (??) propagates along the light cone, −1 3 |g (s,y)| s and supEy∈R Et E decays in time like , while a much faster decay is |Ey| observed whenEy is restricted to the interior of the light cone (i.e., | s | < 1). ii) Because of the support in PE of the vector (t) , the propagation of the elec- ψ j,σt tron current in (??) is limited to the interior of the light cone, up to subleading tails. Combination of i) and ii) is expected to sufﬁce to exhibit decay of the vector norm of (??) and to complete our argument. An important reﬁnement of this reasoning process, involving the term (??), is, however, necessary: A mathematically precise version of statement ii) is as follows: Let χh be a smooth, approximate characteristic function of the support of h. We will prove a propagation estimate

x E σt σt E iγσt (Ev j,∇E ,s) −iE s (t) χh( )e PE e PE ψ s j,σt E σt σt σt iγσt (Ev j,∇E ,s) −iE s (t) − χh(E∇E )e PE e PE ψ PE j,σt 1 1 ≤ | ( )|, ν 3ε ln σt (III.44) s t 2 as s → ∞, where ν > 0 is independent of ε. Using result (I 3) of Theorem ??, and our assumption on the support of h formulated in point ii) of Sect. ??, this estimate provides sufﬁcient control of the asymptotic dynamics of the electron. An important modiﬁcation of the argument above is necessary because of the dependence of Z E 3 Egt (s,Ey) := ΣEv j (Eq) cos(Eq ·Ey − |Eq|s)d q, (III.45) Bκ \Bσt on t, which cannot be neglected even if Ey is in the interior of the light cone. In order to exhibit the desired decay, it is necessary to split Egt (s,Ey) into two pieces, Z E 3 ΣEv j (Eq)cos(Eq ·Ey − |Eq|s)d q (III.46) Bκ \B S σs and Z E 3 ΣEv j (Eq)cos(Eq ·Ey − |Eq|s)d q (III.47) B S \Bσ σs t Infraparticle States in QED I 21

S S −θ for s such that σs > σt , where σs = s , with 0 < θ < 1. (The same procedure will also be used in (??), below.) The function (??) has good decay properties inside the light cone. Expression (??), with Egt (s,Ey) replaced by (??), can be controlled by standard dispersive estimates. The other contribution, proportional to (??), is in principle singular in the infrared region, but is canceled by (??). This can be seen by using a propagation estimate similar to (??). This strategy has been designed in ?. However, because of the vector nature of the interaction in non-relativistic QED, the cancelation in our proof is technically more subtle than the one in ?. After having proven the uniform boundedness of the norms of the approximating vectors ψh,κ (t), one must prove that they deﬁne a Cauchy sequence in H . To this end, we compare these vectors at two different times, t2 > t1 (for the limit t → +∞), and split their difference into

(t2) (t1) (t1) ψh,κ (t1) − ψh,κ (t2) = ∆ψ(t2,σt2 ,G → G ) + ∆ψ(t2 → t1,σt2 ,G ) (t1) +∆ψ(t1,σt2 → σt1 ,G ), (III.48) where the three terms on the r.h.s. correspond to (t ) (t ) I) changing the partition G 2 → G 1 in ψh,κ (t2):

(t2) (t1) ∆ψ(t2,σt2 ,G → G ) N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt2 t2 PE 2 PE 2 1 := e Wκ,σt (Ev j,t2)e e ψ ∑ 2 j,σt2 j=1

N(t ) σ σ 1 E t2 t2 iHt iγσt (Evl( j),∇E ,t2) −iE t2 (t2) − e 2 , (Ev ,t )e 2 PE e PE ψ , ∑ ∑ Wκ σt2 l( j) 2 l( j),σt j=1 l( j) 2 (III.49)

(t2) (t1) (t1) where l( j) labels all cells of G contained in the jth cell G j of G . Moreover,

σ σ E t2 E t1 Evl( j) ≡ ∇EE∗ and Ev j ≡ ∇EE∗ ; Pl( j) Pj

(t ) II) subsequently changing the time, t2 → t1, for the ﬁxed partition G 1 , and

the ﬁxed infrared cut-off σt2 :

(t1) ∆ψ(t2 → t1,σt2 ,G ) N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t2 PE 1 PE 1 1 := e Wκ,σt (Ev j,t1)e e ψ ∑ 2 j,σt2 j=1

N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt2 t2 PE 2 PE 2 1 −e Wκ,σt (Ev j,t2)e e ψ ; (III.50) ∑ 2 j,σt2 j=1

and, ﬁnally, 22 T. Chen, J. Frohlich,¨ A. Pizzo

III) shifting the infrared cut-off, σt2 → σt1 :

(t1) ∆ψ(t1,σt2 → σt1 ,G ) N(t ) σ σ 1 t1 t1 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t1 PE 1 PE 1 1 := e Wκ,σt (Ev j,t1)e e ψ ∑ 1 j,σt1 j=1

N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t2 PE 1 PE 1 1 −e Wκ,σt (Ev j,t1)e e ψ . (III.51) ∑ 2 j,σt2 j=1

It is important to take these three steps in the order indicated above. (t2) (t1) 2 In Step I), the size of k∆ψ(t2,σt2 ,G → G )k in (??) is controlled as follows: The sum of off-diagonal terms yields a subleading contribution. The diagonal terms are shown to tend to 0 by controlling the differences

Evl( j) −Ev j. Infraparticle States in QED I 23

In Step II), Cook’s argument, combined with the cancelation of an infrared tail (as in the mechanism described above), yields the desired decay in t1. Step III) is more involved. But the basic idea is quite simple to grasp: It consists in rewriting

N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t2 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)e e ψ (III.52) ∑ 2 j,σt2 j=1 as

N(t ) σ σ 1 t2 t2 σt σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 2 E 2 t2 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (∇E )Wσt (∇E )e e ψ . ∑ 2 t2 PE 2 PE j,σt2 j=1 (III.53)

(t1) (t1) The term in (??) corresponding to the cell G j of G can then be obtained by acting with the “dressing operator” σt σt −iE 2 t iHt1 ∗ E 2 PE 1 e Wκ,σt (Ev j,t1)W (∇E )e , (III.54) 2 σt2 PE on the “infrared-regular” vector σ σ t2 t2 Z iγσ (Ev j,E∇E ,t1) (t ) iγσ (Ev j,E∇E ,t1) σt 3 e t2 PE Φ 1 := e t2 PE h(PE)Φ 2 d P (III.55) j,σt2 (t1) PE G j corresponding to the vectors Φσ = W (E∇Eσ )Ψ σ (see (??)), for all j. The advan- PE σ PE PE tage of (??) over (??) is that the vector Φ(t1) inherits the regularity properties of j,σt2 Φσ described in Theorem ??. In particular, the vectors Φ(t1) converge strongly, PE j,σt2 as σt2 → 0, and the vector e−iEq·ExΦ(t1) (III.56) j,σt2 depends on Eq in a Holder¨ continuous manner, uniformly in σt2 . This last property entails enough decay to offset various logarithmic divergences appearing in the removal of the infrared cut-off in the dressing operator (??). Our analysis of the strong convergence of the sequence of approximating vectors culminates in the estimate 2 ρ kψh,κ (t2) − ψh,κ (t1)k ≤ O (ln(t2)) /t1 , (III.57) for some ρ > 0. By telescoping, this bound sufﬁces to prove Theorem ??. Indeed, to estimate the difference between the two vectors at times t2 and t1, respectively, 2 n where t2 > t1 > 1, we may consider a sequence of times {t1 ,...,t1 }, such that n n+1 t1 ≤ t2 < t1 , and use Estimate (??) for each difference n ψh,κ (t2) − ψh,κ (t1 ), (III.58) m m−1 ψh,κ (t1 ) − ψh,κ (t1 ), 2 ≤ m ≤ n. (III.59) Then, one can show that there exists a constant ρ0 > 0 such that the rate of conver- 0 gence of the time-dependent vector is at least of order t−ρ , as stated in Theorem ??. 24 T. Chen, J. Frohlich,¨ A. Pizzo

III.5 A space of scattering states

(out/in) ˚ out/in We use the asymptotic states ψh,κ to construct a subspace, Hκ , of scattering states invariant under space-time translations, and with a photon energy threshold κ, n o ˚ out/in _ out/in E 1 3 Hκ := ψh,κ (τ,Ea) : h(P) ∈ C0(S \Brα ), τ ∈ R, Ea ∈ R , (III.60) where

out/in −iEa·PE −iHτ out ψh,κ (τ,Ea) ≡ e e ψh,κ . (III.61) This space contains states describing an asymptotically freely moving electron, accompanied by asymptotic free photons with energy smaller than κ. ˚ out/in Spaces of scattering states are obtained from the space Hκ by adding (and subtracting) “hard” photons, i.e.,

n_ out/in o out/in := ψ : h(PE) ∈ C1( \ ), FEˆ ∈ C∞( 3\0; 3) , (III.62) H h,FE 0 S Brα 0 R C where out/in i AE[FEt ,t]−AE[FEt ,t] ψ := s − lim e ψh,κ (t), (III.63) h,FE t→+/−∞ and Z ! ∂FEt (Ey) ∂AE(t,Ey) 3 AE[FEt ,t] := i AE(t,Ey) · − · FEt (Ey) d y (III.64) ∂t ∂t is the L.S.Z. photon ﬁeld smeared out with the vector test function Z 3 d k ∗ λ −i|k|t+iEk·Ey FEt (Ey) := Eε Fˆ (Ek)e (III.65) ∑ 3 p Ek,λ λ=± (2π) 2 |k| with

FEˆ (Ek) := Eε ∗ Fˆ λ (Ek) ∈ C∞( 3\{0}; 3). (III.66) ∑ Ek,λ 0 R C λ An a posteriori physical interpretation of the scattering states constructed here emerges by studying how certain algebras of asymptotic operators are represented on the spaces of scattering states: out/in • The Weyl algebra, Aph , associated with the asymptotic electromagnetic ﬁeld. out/in • The algebra Ael generated by smooth functions of compact support of the asymptotic velocity of the electron. These algebras will be deﬁned in terms of the limits (??) and (??), below, whose existence is established in Sect. ??. Infraparticle States in QED I 25

∞ 3 iHt Ex −iHt Theorem III3 Functions f ∈ C0 (R ), of the variable e t e , have strong limits, as t → ±∞, as operators acting on H out/in,

Ex out/in out/in s − lim eiHt f ( )e−iHt ψ =: ψ , (III.67) h,FE f h,FE t→+/−∞ t E∇E where f (PE) := lim f (E∇Eσ ). E∇E σ→0 PE Theorem III4 The LSZ Weyl operators i AE[GEt ,t]−AE[GEt ,t] λ 2 3 −1 3 e : Gˆ (Ek) ∈ L (R ,(1 + |Ek| )d k), λ = ± , (III.68) have strong limits in H out/in; i.e., i AE[GE ,t]−AE[GE ,t] W out/in(GE) := s − lim e t t (III.69) t→+/−∞ exists. The limiting operators are unitary and satisfy the following properties: i)

E E0 out/in out/in 0 out/in 0 − ρ(G,G ) W (GE)W (GE ) = W (GE + GE )e 2 , (III.70) where ! Z λ ρ(GE,GE0) = 2iIm ∑ Gˆλ (Ek)Gˆ0 (Ek)d3k . (III.71) λ

out/in ii) The mapping R 3 s −→ W (sGE) deﬁnes a strongly continuous one- parameter group of unitary operators. iii)

iHτ out/in −iHτ out/in e W (GE)e = W (GE−τ ), (III.72)

where GE−τ is the freely time-evolved (vector) test function at time −τ. out/in out/in The two algebras, Aph and Ael , commute. This is the precise mathematical expression of the asymptotic decoupling of the dynamics of photons from the one of the electron. The proof is non-trivial, because non-Fock representations of the asymptotic photon creation- and annihilation operators appear. (For the represen- out/in tation of Aph , which is non-Fock but locally Fock see Sect. ??.) We will show that

out/in More precisely, the representation of Aph on the space of scattering states can be decomposed in a direct integral of inequivalent irreducible representations labelled by the asymptotic velocity of the electron. For different values of the asymptotic velocity, these representations turn out to be inequivalent. Only for a vanishing electron velocity, the representation is Fock; for non-zero velocity, it is a coherent non-Fock representation. The coherent photon cloud, labeled by the asymptotic velocity, is the one anticipated by Bloch and Nordsieck in the non- relativistic approximation. These results can be interpreted as follows: In every scattering state, an asymptotically freely moving electron is observed (with an asymptotic velocity whose size is strictly smaller than the speed of light, by construction) accompanied by a cloud of asymptotic photons propagating along the light cone.

Remark We point out that, in our deﬁnition of scattering states, we can directly accommodate an arbitrarily large number of “hard” photons without energy restriction, i.e., we can construct the limiting vector

out (m) out (1) out (m) (1) AE [FE ]...AE [FE ]ψ := s − lim AE[FEt ,t]...AE[FEt ,t]ψh,κ (t) h,κ t→+∞ (III.76)

out which represents the state ψh,κ plus m asymptotic photons with wave functions FE(m),...,FE(1), respectively. Analogously, we deﬁne

in (m0) in (1) in (m0) (1) AE [GE ]...AE [GE ]ψ := s − lim AE[GEt ,t]...AE[GEt ,t]ψh,κ (t). h,κ t→−∞ (III.77)

This is possible because, apart from some higher order estimates to control the commutator i[H, Ex], and the photon creation operators in (??)–(??) (see for example ?), we use the propagation estimate (??), which only limits the asymptotic velocity of the electron. This fact is very important for estimating scattering amplitudes involving an arbitrary number of “hard” photons. 0 In particular, for any m,m ∈ N, we can deﬁne the S-matrix element 0 m,m E E Eout E(m) Eout E(1) out Sα ({Fi}, {G j}) = A [F ]...A [F ]ψhout ,κout , 0 Ein E(m ) Ein E(1) in A [G ]...A [G ]ψhin,κin (III.78) which corresponds to the transition amplitude between two states describing an incoming electron with wave function hin, accompanied by a soft photon cloud of free photons of energy smaller than κin, plus m0 hard photons (with wave functions 0 GE(1),...,GE(m )), and an outgoing electron with wave function hout and soft photon energy threshold κout , plus m hard photons, respectively. Infraparticle States in QED I 27

m,m0 The expansion of Sα ({FEi}, {GE j}) in the finestructure constant α can be carried out, at least to leading order, along the lines of ?. This yields a rigorous proof of the transition amplitudes for Compton scattering in leading order, and in the non-relativistic approximation, that one can find in textbooks. Moreover, as expected from classical electromagnetism, “close” to the electron a Lienard-Wiechert´ electromagnetic field is observed. The precise mathematical statement is Z lim lim |dE|2 ψout/in, eiHt d3y F (0,y)δ˜ (y − x − dE)e−iHt ψout/in |dE|→∞ t→±∞ h,FE µν E Λ E E h,FE

Z E∇E − F PE (0,dE)hψσt , ψσt i|h(PE)|2d3P = 0, (III.79) µν PE PE ˜ where δΛ is a smooth, Λ-dependent delta function which has the property that its Fourier transform is supported in BΛ , Ex is the electron position,

Fµν = ∂µ Aν − ∂ν Aµ (III.80) with (2π)2α1/2 A0(0,Ey) := − (III.81) |Ey −Ex| Z d3k A (0,y) := − (Eε )i e−iEk·Eya∗ + (Eε ∗ )i eiEk·Eya , (III.82) i E ∑ q Ek,λ Ek,λ Ek,λ Ek,λ λ |Ek|

E ∇EPE and Fµν is the electromagnetic ﬁeld tensor corresponding to a Lienard-Wiechert´ solution for the current 1 1 3 2 (3) E 3 2 E (3) E E Jµ (t,Ey):= −2(2π) α δ (Ey−∇EPEt),2(2π) α ∇EPEδ (Ey−∇EPEt) , |∇EPE|<1; (III.83) see the discussion in Sect. ??.

IV Uniform Boundedness of the Limiting Norm

Our first aim is to prove the uniform boundedness of kψh,κ (t)k, as t → ∞; more precisely, that Z 2 3 lim ψh,κ (t), ψh,κ (t) = |h(PE)| d P. (IV.1) t→∞ The sum of the diagonal terms – with respect to the partition G (t) introduced above – is easily seen to yield R |h(PE)|2d3P in the limit t → ∞, as one expects. Thus, our main task is to show that the sum of the off-diagonal terms vanishes in this limit. In Sect. ??, we prove that the norm-bounded sequence {ψh,κ (t)} is, in fact, Cauchy. We recall that the definition of the vector ψh,κ (t) involves three different rates: −β • The rate t related to the fast infrared cut-off σt ; −θ S • the rate t of the slow infrared cut-off σt (see (??)); • the rate t−ε of refinement of the cell partitions G (t). 28 T. Chen, J. Frohlich,¨ A. Pizzo

IV.1 Control of the off-diagonal terms

We denote the off-diagonal term labeled by the pair (l, j) of cell indices l 6= j contributing to the l.h.s. of (??) by

σ σ σ σ D iγ (Ev ,∇E t ,t) −iE t t (t) iγ (Ev ,∇E t ,t) −iE t t (t) E M (t) := e σt l PE e PE ψ , (t)e σt j PE e PE ψ , l, j l,σt Wκ,σt ,l, j j,σt (IV.2) where we use the notation ! Z n o (t) := exp Eη (Ek) · Eε a∗ e−i|Ek|t −Eε ∗ a ei|Ek|t d3k , Wκ,σt ,l, j ∑ l, j Ek,λ Ek,λ Ek,λ Ek,λ λ Bκ \Bσt (IV.3)

(t) E σt ∗ ∗ where Ev j ≡ ∇E (PEj ), PEj being the center of the cell G j , and

1 Ev j 1 Evl E 2 2 Eηl, j(k) := α 3 − α 3 . (IV.4) |Ek| 2 (1 −bk ·Ev j) |Ek| 2 (1 −bk ·Evl)

We study the limit t → +∞; the case t → −∞ is analogous. A. Asymptotic orthogonality. In order to prove that the off-diagonal terms (IV.2) vanish in the limit t → +∞, we separate the role played by the time variable t as the parameter determining the dynamical cell decomposition and infrared cutoffs, from its usual role as the conjugate variable to the energy. For the latter, we introduce an auxiliary variable s ≥ t. Then, for fixed t (such that the cell decomposition and the cutoffs are constant), we interpret the terms (IV.2) as special values 1 µ Ml, j(t) = Ml, j(t,t) of families Ml, j(t,s) introduced below, which depend on t,s, and an additional auxiliary parameter µ ∈ . Our strategy will be based on prov- µ R ing that the dispersive properties of Ml, j(t,s) as a function of s ≥ t alone, for fixed t and µ, imply that Ml, j(t) has a sufficiently fast decay in t such that our desired result of asymptotic orthogonality follows. More precisely, we introduce a family of operators ! Z n o µ (s) := exp µ Eη (Ek) · Eε a∗ e−i|Ek|s −Eε ∗ a ei|Ek|s d3k Wκ,σt ,l, j ∑ l, j Ek,λ Ek,λ Ek,λ Ek,λ λ Bκ \Bσt (IV.5) depending on a parameter µ ∈ R, and define

σ σ σ σ µ D iγ (Ev ,∇E t ,s) −iE t s (t) µ iγ (Ev ,∇E t ,s) −iE t s (t) E M (t,s) := e σt l PE e PE ψ , (s)e σt j PE e PE ψ bl, j l,σt Wκ,σt ,l, j j,σt (IV.6)

µ=1 for s ≥ t( 1). Obviously, Mbl, j (t,t) = Ml, j(t). Infraparticle States in QED I 29

The phase factor γ (v ,∇Eσt ,s) is chosen as follows: σt Ej PE Z s Z γ (v ,∇Eσt ,s) := −α E∇Eσt · ΣE (Ek)cos(Ek ·E∇Eσt τ − |Ek|τ)d3kdτ σt Ej PE PE Ev j PE 1 B S \Bσt στ (IV.7)

−θ for s ≥ σt , and

− 1 θ Z σt Z γ (v ,∇Eσt ,s) := −α E∇Eσt · ΣE (Ek)cos(Ek ·E∇Eσt τ − |Ek|τ)d3kdτ, σt Ej PE PE Ev j PE 1 B S \Bσt στ (IV.8)

−θ for s < σt . As a function of µ, the scalar product in (??) satisﬁes the ordinary differential equation

µ dMbl, j(t,s) µ µ = − µ Cl, j,σ Mb (t,s) + rσ (t,s), (IV.9) dµ t l, j t where Z E 2 3 Cl, j,σt := |Eηl, j(k)| d k, (IV.10) Bκ \Bσt and

σ σ µ D iγ (Ev ,E∇E t ,s) −iE t s (t) r (t,s) := − e σt l PE e PE ψ , σt l,σt σ σ µ iγ (Ev ,E∇E t ,s) −iE t s (t) E (s)a (Eη )(s)e σt j PE e PE ψ Wκ,σt ,l, j Eσt l, j j,σt σ σ D iγ (Ev ,E∇E t ,s) −iE t s (t) + a (Eη )(s)e σt l PE e PE ψ , Eσt l, j l,σt σ σ µ iγ (Ev ,E∇E t ,s) −iE t s (t) E (s)e σt j PE e PE ψ , Wκ,σt ,l, j j,σt with Z a (Eη )(s) := Eη (Ek) ·Eε ∗ a ei|Ek|s d3k. (IV.11) Eσt l, j ∑ l, j Ek,λ Ek,λ λ Bκ \Bσt The solution of the ODE (??) is given by

C Z µ C µ l, j,σt 2 µ0 l, j,σt 2 02 − 2 µ 0 − 2 (µ −µ ) 0 Mbl, j(t,s) = e Mbl, j(t,s) + rσt (t,s)e dµ , (IV.12) 0 where the initial condition at µ = 0 is given by

0 Mbl, j(t,s) = 0, (IV.13) since the supports in PE of the two vectors ψ(t) , ψ(t) are disjoint (up to sets of l,σt j,σt measure 0), for arbitrary t and s. 30 T. Chen, J. Frohlich,¨ A. Pizzo

Furthermore, condition ( 4) in Theorem ?? implies that the vectors ψ(t) , I l,σt (t) are vacua for the annihilation part of the asymptotic photon field1 under the ψ j,σt dynamics generated by the Hamiltonian Hσt . As a consequence, we find that µ lim rσ (t,s) = 0, (IV.14) s→+∞ t for fixed µ and t. To arrive at this conclusion, the following is used: The one- σ iγ (Ev ,E∇E t ,s) particle state, multiplied by the phase e σt j PE continues to be a one-particle state for the Hamiltonian Hσt ; for large s (see (??)) the phase is s-independent; the operator Eσt coincides with the operator Hσt when applied to one-particle states PE of the Hamiltonian Hσt . Therefore, by dominated convergence, it follows that

1 lim Mb (t,s) = 0. (IV.15) s→+∞ l, j

1 Since Mbl, j(t,t) ≡ Ml, j(t), we have

|Ml, j(t)| 1 = |Ml, j(t) − Mbl, j(t,+∞)| Z +∞ σt σt (t) d iHσt s iγ (Ev j,∇E ,s) −iE s (t) ≤ 2sup(kψ k)sup k e (v ,s)e σt PE e PE ψ dsk . l,σt Wκ,σt Ej j,σt l j t ds (IV.16) To estimate the r.h.s. of (??), we proceed as follows. Since we are interested in the limit t → +∞, and the integration domain on the r.h.s. of (??) is [t,+∞), our aim is to show that

d σ σt σt iH t s iγσt (Ev j,∇E ,s) −iE s (t) e W , (Ev j,s)e PE e PE ψ (IV.17) ds κ σt j,σt is integrable in s, and that the rate at which the r.h.s. of (??) converges to zero offsets the growth of the number of cells in the partition. This allows us to conclude that

N(t) ∑ Ml, j(t) −→ 0 (IV.18) l, j (l6= j) in the limit t → +∞, and, as a corollary,

N(t) Z 2 3 lim Ml, j(t) = |h(PE)| d P, (IV.19) t→+∞ ∑ l, j as asserted in Theorem ??. The convergence (??) follows from the following theorem. 1 The existence of the asymptotic ﬁeld operator for a ﬁxed cut-off dynamics is derived as explained in part B. of this section. Infraparticle States in QED I 31

Theorem IV1 The off-diagonal terms Ml, j(t), l 6= j, satisfy

1 |M (t)| ≤ C |ln σ |2 t−3ε , (IV.20) l, j tη t for some constants C < ∞ and η > 0, both independent of l, j, and ε > 0. In particular, Ml, j(t) → 0, as t → +∞.

As a corollary, we ﬁnd

1 1 |M (t)| ≤ CN2(t) |ln σ |2 t−3ε ≤ C0 |ln σ |2 t3ε , (IV.21) ∑ l, j tη t tη t 1≤l6= j≤N(t) since N(t) ≈ t3ε . (Throughout the paper, C,C0,c, and c0 denote positive constants.) η We conclude that, for ε < 4 ,(??) follows. B. Time derivative and infrared tail. We now proceed to prove Theorem ??. The arguments developed here will also be relevant for the proof of the (strong) convergence of the vectors ψh,κ (t), as t → +∞, which we discuss in Sect. ??. To control (??), we focus on the derivative

d σ E σt σt iH t s iγσt (Ev j,∇E ,s) −iE s (t) e W , (Ev j,s)e PE e PE ψ (IV.22) ds κ σt j,σt σt σt iHσt s σ iγ (Ev j,E∇E ,s) −iE s (t) = ie [H t , (v ,s)]e σt PE e PE (IV.23) I Wκ,σt Ej ψ j,σt E σt σ dγσt (Ev j,∇E ,s) E σt σt iH t s PE iγσt (Ev j,∇E ,s) −iE s (t) +ie W , (Ev j,s) e PE e PE ψ , (IV.24) κ σt ds j,σt where

σt σt σ 1 AE (Ex) · AE (Ex) H t := α 2 Ep · AEσt (Ex) + α . (IV.25) I 2

−isH0 isH0 σt σt We have used that Wκ,σt (Ev j,s) = e Wκ,σt (Ev j,0)e , where H0 := H − HI is the free Hamiltonian, to obtain the commutator in (??). We rewrite the latter in the form

σt ∗ σt σt [HI , Wκ,σt (Ev j,s)] = Wκ,σt (Ev j,s) Wκ,σt (Ev j,s) HI Wκ,σt (Ev j,s) − HI , (IV.26) and use that

1 Z ∗Eσt Eσt 2 E E E E 3 Wκ,σt (Ev j,s) A (Ex)Wκ,σt (Ev j,s) = A (Ex) + α ΣEv j (k)cos(k ·Ex − |k|s)d k Bκ \Bσt (IV.27) 32 T. Chen, J. Frohlich,¨ A. Pizzo

(see (??)), where Σ l (Ek) is deﬁned in (??). We can then write the term (??) as Ev j

σ Z iH t s σt 3 E E (??) = ie Wκ,σt (Ev j,s)α i[H ,Ex] · d kΣEv j (k) (IV.28) Bκ \Bσt σ σ −iE t s iγ (Ev ,E∇E t ,s) (t) ×cos(Ek · x − |Ek|s) e PE e σt j PE E ψ j,σt 2 Z 2 iHσt s α E E E E 3 + ie Wκ,σt (Ev j,s) ΣEv j (k)cos(k ·Ex − |k|s)d k (IV.29) 2 Bκ \Bσt σ σ iγ (Ev ,E∇E t ,s) −iE t s (t) ×e σt j PE e PE , ψ j,σt

1 where we recall that i[Hσt ,Ex] = Ep + α 2 AEσt (Ex); see (??). From the decay estimates provided by Lemma ?? in the Appendix one con- cludes that the norm of (??) is integrable in s, and that Z +∞ 1 2 − 3ε dsk(??)k ≤ | ln σt | t 2 (t 1) (IV.30) t tη for some η > 0 independent of ε. The analysis of (??) is more involved. Our argument will eventually involve the derivative of the phase factor in (??). To begin with, we write (??) as

σ s iH t σt 1 (??) = ie Wκ,σ (Ev j,s)α i[H ,Ex] · t Hσt + i Z σt σt σ 3 −iE s iγσ (Ev j,E∇E ,s) (t) ×(H t + i) d k E (Ek)cos(Ek · x − |Ek|s) e PE e t PE . ΣEv j E ψ j,σt Bκ \Bσt (IV.31)

Pulling the operator (Hσt + i) through to the right, the vector (??) splits into the sum of a term involving the commutator [Hσt ,Ex],

σ s Z iH t σt 1 3 ie W , (Ev j,s)α i[H ,Ex] · d kΣEv (Ek) κ σt σt Ej H + i Bκ \Bσt σt σt σ iγσ (Ev j,E∇E ,s) −iE s (t) ×[H t , cos(Ek · x − |Ek|s)] e t PE e PE , (IV.32) E ψ j,σt and

σ Z iH t s σt 3 E E ie Wκ,σt (Ev j,s)α i[H ,Ex] · d kΣEv j (k) Bκ \Bσt σ σ 1 iγ (Ev ,E∇E t ,s) −iE t s σ (t) × cos(Ek ·Ex − |Ek|s)e σt j PE e PE (E t + i)ψ . (IV.33) Hσt + i PE j,σt

σt 1 Note that [H ,Ex] Hσt +i is bounded in the operator norm, uniformly in σt . To control (??) and (??), we invoke a propagation estimate for the electron position operator as follows. Due to condition (I 3) in Theorem ??, we can introduce a ∞ 3 C −function χh(Ey), Ey ∈ R , such that

• χh(Ey) = 1 for νmin ≤ |Ey| ≤ νmax. 1 1+νmax • χh(Ey) = 0 for |Ey| ≤ 2 νmin and |Ey| ≥ 2 . Infraparticle States in QED I 33

It is shown in Theorem ?? of the Appendix that, for θ < 1 sufﬁciently close to 1 and s large, the propagation estimate

x E σt σt E σt σt E iγσt (Ev j,∇E ,s) −iE s (t) σt iγσt (Ev j,∇E ,s) −iE s (t) χh e PE e PE ψ − χh(E∇E )e PE e PE ψ s j,σt PE j,σt 1 1 ≤ | ( )|, ν 3ε ln σt (IV.34) s t 2 holds, where ν > 0 is independent of ε. The argument uses the Holder¨ regularity of E∇Eσ and of Φσ listed under properties ( 2) in Theorem ??, differentiability PE PE I of h(PE), and (??). We continue with the discussion of the expressions (??) and (??). We split (??) into two parts:

σ σ σt i (v ,E E t ,s) −iE t s (t) iH s κ γσt Ej ∇ E E σt ie Wκ,σ (Ev j,s)J | S (s)e P e P (E + i)ψ (IV.35) t σs PE j,σt S σt σt iHσt s σ iγ (Ev j,E∇E ,s) −iE s σ (t) + ie (v ,s) | s (s) e σt PE e PE (E t + i)ψ , (IV.36) Wκ,σt Ej J σt PE j,σt using the deﬁnitions

S Z σs σt 1 3 −θ J | (s) := α i[H ,Ex] · ΣEv (Ek) cos(Ek ·Ex − |Ek|s)d k if s ≥ σt σt σt Ej H + i B S \Bσ σs t −θ := 0 if s < σt , (IV.38)

S −θ where we refer to σs := s as the slow infrared cut-off. (We consider s,t large S enough such that κ > σs ,σt .) S σs To control J |σt (s) in (??), we deﬁne the “infrared tail”

Ex iHσt s −iHσt s dγσ (Ev j, ,s) σt 1 d(e Ex (s)e ) σt b t s := α e−iH s h eiH s · ds Hσt + i ds Z · ΣE (Ek) cos(Ek ·E∇Eσt s − |Ek|s)d3k if s−θ ≥ σ , Ev j PE t B S \Bσ σs t −θ := 0 if s < σt , (IV.39) 34 T. Chen, J. Frohlich,¨ A. Pizzo

Ex where Exh(s) := Exχh( s ). Summarizing, we can write (??) as (??) = (??) σ σ σt i (v ,E E t ,s) −iE t s (t) iH s κ γσt Ej ∇ E E σt +ie Wκ,σ (Ev j,s)J | S (s)e P e P (E + i)ψ (IV.40) t σs PE j,σt S σt σt iHσt s σ iγ (Ev j,E∇E ,s) −iE s σ (t) +ie (v ,s) | s (s) e σt PE e PE (E t + i)ψ (IV.41) Wκ,σt Ej J σt PE j,σt E σt σ dγσt (Ev j,∇E ,s) σt σt iH t s PE iγσt (Ev j,∇E ,s) −iE s (t) +ie W , (Ev j,s) e PE e PE ψ (IV.42) κ σt ds j,σt σ iH t s σt 1 +ie Wκ,σ (Ev j,s)α i[H ,Ex] · (IV.43) t Hσt + i Z σt σt 3 σ iγσ (Ev j,E∇E ,s) −iE s (t) × d k E (Ek)[H t ,cos(Ek · x − |Ek|s)]e t PE e PE , ΣEv j E ψ j,σt Bκ \Bσt where we recall that (??) satisﬁes (??). We claim that

Z +∞ 1 2 −3ε/2 [(??) + (??) + (??)]ds ≤ |ln σt | t ,(t 1) (IV.44) t tη for some η > 0 depending on θ, but independent of ε. This is obtained from

Z +∞ [(??) + (??) + (??)]ds (IV.45) t Z +∞ Z σt 1 3 ≤ ds α i[H ,Ex] · d kΣEv (Ek) cos(Ek ·Ex − |Ek|s) σt Ej t H + i Bκ \Bσt x E σt σt σt E iγσt (Ev j,∇E ,s) −iE s σt (t) × χh(E∇E ) − χh( ) e PE e PE (E + i)ψ (IV.46) PE s PE j,σt Z +∞ σt σt κ Ex iγσ (Ev j,E∇E ,s) −iE s σ (t) + ds | (s)χ ( )e t PE e PE (E t + i)ψ (IV.47) J σ S h E j,σt t s s P +∞ " Ex # Z σt S Ex dγσ (Ev j, ,s) + dseiH s (v ,s) |σs (s) ( ) − b t s (IV.48) Wκ,σt Ej J σt χh t s ds

σ σ iγ (Ev ,E∇E t ,s) −iE t s σ (t) ×e σt j PE e PE (E t + i)ψ PE j,σt

σ +∞ " E t Ex # Z σ dγσt (Ev j,∇E ,s) dγ (Ev j, ,s) + dseiH t s (v ,s) PE − bσt s Wκ,σt Ej (IV.49) t ds ds

σ σ iγ (Ev ,E∇E t ,s) −iE t s σ (t) ×e σt j PE e PE (E t + i)ψ PE j,σt using the following arguments:

1 2 −3ε/2 • The term (??) can be bounded from above by tη |ln σt | t , for some η > 0 independent of ε, due to the propagation estimate for (??) and Lemma ??, which show that the integrand has a sufﬁciently strong decay in s. Infraparticle States in QED I 35

S Ex • In (??), the slow cut-off σs and the function χh( s ) make the norm integrable in s with the desired rate (i.e., to get a bound as in (??)), for a suitable choice of θ < 1. In particular, we can exploit that

Z Ex sθ sup d3k E(Ek,v )cos(Ek · x − |Ek|s) ( ) ≤ ( ), (IV.50) Σ Ej E χh O 2 3 Bκ \B S s s Ex∈R σs see Lemma ?? in the Appendix. • In (??), only terms integrable in s and decaying fast enough to satisfy the bound (??) are left after subtracting

Ex dγσ (Ev j, ,s) b t s (IV.51) ds

S σs from J |σt (s). This is explained in detail in the proof of Theorem ?? in the Appendix. • To bound (??), we use the electron propagation estimate, combined with an integration by parts, to show that the derivative of the phase factor tends to the “infrared tail” for large s, at an integrable rate that provides a bound as in (??). We note that due to the vector interaction in non-relativistic QED, this argument is more complicated here than in the Nelson model treated in ? where the interaction term is scalar. Here, we have to show (see Theorem ??) that, in σ σ the integral with respect to s, the pointwise velocity eiH t si[Ex,Hσt ]e−iH t s can be replaced by the (asymptotic) mean velocity E∇Eσt at asymptotic times. PE Finally, to control (??), we observe that the commutator introduces additional Ex decay in s into the integrand when s is restricted to the support of χh. It then follows that the propagation estimate sufﬁces (without infrared tail) to control the norm, by the same arguments that were used to estimate (??), (??). Combining the above arguments, the proof of Theorem ?? is completed.

V Proof of Convergence of ψh,κ (t)

In this section, we prove that ψh,κ (t) deﬁnes a bounded Cauchy sequence in H , as t → +∞. To this end, it is necessary to control the norm difference between vectors ψh,κ (ti), i = 1,2, at times t2 > t1 1.

V.1 Three key steps

As anticipated in Sect. ??, we decompose the difference of ψh,κ (t1) and ψh,κ (t2) into three terms

(t2) (t1) (t1) ψh,κ (t1) − ψh,κ (t2) = ∆ψ(t2,σt2 ,G → G ) + ∆ψ(t2 → t1,σt2 ,G ) (t1) +∆ψ(t1,σt2 → σt1 ,G ), (V.1) where we recall from (??)–(??): 36 T. Chen, J. Frohlich,¨ A. Pizzo

I) The term

(t2) (t1) ∆ψ(t2,σt2 ,G → G ) N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt2 t2 PE 2 PE 2 1 = e Wκ,σt (Ev j,t2)e e ψ ∑ 2 j,σt2 j=1

N(t ) σ σ 1 E t2 t2 iHt iγσt (Evl( j),∇E ,t2) −iE t2 (t2) − e 2 , (Ev ,t )e 2 PE e PE ψ , (V.2) ∑ ∑ Wκ σt2 l( j) 2 l( j),σt j=1 l( j) 2

(t ) (t ) accounts for the change of the partition G 2 → G 1 in ψh,κ (t2), where (t2) (t1) (t1) l( j) labels the sub-cells belonging to the sub-partition G ∩G j of G j , and where we deﬁne σ σ E t2 E t1 Evl( j) ≡ ∇EE∗ and Ev j ≡ ∇EE∗ ; Pl( j) Pj II) the term

(t1) ∆ψ(t2 → t1,σt2 ,G ) N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t2 PE 1 PE 1 1 = e Wκ,σt (Ev j,t1)e e ψ ∑ 2 j,σt2 j=1

N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt2 t2 PE 2 PE 2 1 − e Wκ,σt (Ev j,t2)e e ψ , (V.3) ∑ 2 j,σt2 j=1

accounts for the subsequent change of the time variable, t2 → t1, for the (t1) fixed partition G , and the fixed infrared cut-off σt2 ; and finally, III) the term

(t1) ∆ψ(t1,σt2 → σt1 ,G ) N(t ) σ σ 1 t1 t1 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t1 PE 1 PE 1 1 = e Wκ,σt (Ev j,t1)e e ψ ∑ 1 j,σt1 j=1

N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t2 PE 1 PE 1 1 − e Wκ,σt (Ev j,t1)e e ψ (V.4) ∑ 2 j,σt2 j=1

accounts for the change of the infrared cut-off, σt2 → σt1 . Our goal is to prove 2 ρ kψh,κ (t2) − ψh,κ (t1)k ≤ O (ln(t2)) /t1 , (V.5) for some ρ > 0. To this end, it is necessary to perform the three steps in the order displayed above. As a corollary of the bound (??), we obtain Theorem ?? by telescoping (see the comment after Eq. (??)). The arguments in our proof are very similar to those in ?, but a number of modiﬁcations are necessary because of the vector nature of the QED interaction. For these modiﬁcations, we provide detailed explanations. Infraparticle States in QED I 37

V.2 Reﬁning the cell partition

In this section, we discuss step (??) in which the momentum space cell partition is modiﬁed. It is possible to apply the methods developed in ?, up to some minor modiﬁcations. We will prove that

(t2) (t1) 2 ρ ∆ψ(t2,σt2 ,G → G ) ≤ O (ln(t2)) /t1 (V.6) for some ρ > 0. The contributions from the off-diagonal terms with respect to the sub-partition G (t2) of G (t1) can be estimated by the same arguments that have culminated in the proof of Theorem ??. That is, we ﬁrst express ψ(t1) as j,σt2

Z σ Z σ (t1) E t2 3 E t2 3 (t2) ψ j,σ = (t ) h(P)Ψ d P = (t ) h(P)Ψ d P = ψ . (V.7) t2 1 PE ∑ 2 PE ∑ l( j),σt2 G j l( j) Gl( j) l( j)

Then,

(t2) (t1) 2 ∆ψ(t2,σt 2,G → G )

N(t ) σ 1 t2 D h i −iE t2 (t2) = , (Ev ,t ) − , (Ev j,t ) e PE ψ , ∑ ∑ Wcκ σt2 l( j) 2 Wcκ σt2 2 l( j),σt j, j0=1 l( j),l0( j0) 2

σt h i −iE 2 t (t ) E PE 2 2 Wcκ,σt (Evl0( j0),t2) − Wcκ,σt (Ev j0 ,t2) e ψ 0 0 , (V.8) 2 2 l ( j ),σt2 where we deﬁne

σ t2 iγσ (Ev j,E∇E ,t ) (v ,t ) := (v ,t )e t2 PE 2 , (V.9) Wcκ,σt2 Ej 2 Wκ,σt2 Ej 2 σ t2 iγσ (Ev ,E∇E ,t ) (v ,t ) := (v ,t )e t2 l( j) PE 2 . (V.10) Wcκ,σt2 El( j) 2 Wκ,σt2 El( j) 2 Following the analysis in Sect. ??, one ﬁnds that the sum over pairs (l0( j0), l( j)) 0 0 −ε η with either l 6= l or j 6= j can be bounded by O(t2 ), provided that ε < 4 , as in (??). Let h i stand for the expectation value with respect to the vector Ψ. Then, Ψe e we are left with the diagonal terms

N(t1) Dh ∗ ∗ iE 2 − Wcκ,σ (Evl( j),t2)Wcκ,σt (Ev j,t2) − Wcκ,σ (Ev j,t2)Wcκ,σt (Evl( j),t2) , ∑ ∑ t2 2 t2 2 Ψ j=1 l( j) e (V.11)

σt −iE 2 t (t ) labeled by pairs (l( j), l( j)), where Ψe ≡ e PE 2 ψ 2 in the case considered l( j),σt2 here. For each term

D ∗ E Wcκ,σt (Evl( j),t2)Wcκ,σt (Ev j,t2) , (V.12) 2 2 Ψe 38 T. Chen, J. Frohlich,¨ A. Pizzo we can again invoke the arguments developed for off-diagonal elements indexed by (l, j) (where l 6= j) from Sect. ??. In particular, we deﬁne for s > t2,

µ D −iEσt s (t ) ∗ µ PE 2 Mb (t2,s) := e ψ , Wcκ,σt (Ev j,s) [l( j),Ev j],[l( j),Evl( j)] l( j),σt2 2 µ −iEσt s (t ) E PE 2 Wcκ,σt (Evl( j),s)e ψ , (V.13) 2 l( j),σt2

where

σ E t2 µ∗ µ −iγσt (Ev j,∇E ,s) µ ∗ W (Ev j,s)W (Ev ,s) = e 2 PE W (Ev j,s) cκ,σt2 cκ,σt2 l( j) κ,σt2 σ t2 µ iγσ (Ev ,E∇E ,s) ×W (Ev ,s) e t2 l( j) PE , (V.14) κ,σt2 l( j)

and

Z µ ∗ µ n ∗ −i|Ek|s W (Ev j,s)W (Ev ,s) = exp µ Eη (Ek) · Eε a e κ,σt2 κ,σt2 l( j) ∑ j,l( j) Ek,λ Ek,λ Bκ \Bσ λ t2 ! o −Eε ∗ a ei|Ek|s d3k , (V.15) Ek,λ Ek,λ

with µ a real parameter. Infraparticle States in QED I 39

Proceeding similarly as in (??), the solution of the ODE analogous to (??) for µ Mb (t2,s) at µ = 1 consists of a contribution at µ = 0, which remains [l( j),Ev j],[l( j),Evl( j)] non-zero as s → ∞, and a remainder term that vanishes in the limit s → ∞. In fact, 1 lim Mb (t2,s) s→+∞ [l( j),Ev j],[l( j),Evl( j)] 1 1 C σ σ − σ − j,l( j) t2 t2 θ t2 θ − D iγσ (Ev j,E∇E ,σ ) (t ) iγσ (Ev ,E∇E ,σ ) (t ) E = e 2 e t2 PE t2 ψ 2 , e t2 l( j) PE t2 ψ 2 , (V.16) l( j),σt2 l( j),σt2 where Z C := |Eη (Ek)|2d3k, (V.17) j,l( j),σt2 j,l( j) Bκ \Bσ t2 as in (??), with Eη j,l( j)(Ek) deﬁned in (??). Hence, (??) is given by the sum of

N(t2) Z +∞ d 1 − M (t2,s)ds ∑ ∑ b[`( j),Ev j],[`( j),Ev`( j)] j=1 `( j) t2 ds

N(t2) Z +∞ d 1 − M (t2,s)ds (V.18) ∑ ∑ b[`( j),Ev`( j)],[`( j),Ev j] j=1 `( j) t2 ds and N t " C # ( 2) l( j), j,σt D (t2) σt2 − 2 (t2) E ψ , 2 − 2cos ∆γ (Ev j −Ev ,E∇E ,t2) e 2 ψ , ∑ ∑ l( j),σt σt2 l( j) PE l( j),σt j=1 `( j) 2 2 (V.19) where 1 1 σt2 σt2 − θ σt2 − θ ∆γ (Ev j −Ev ,E∇E ,t2) := γ (Ev ,E∇E ,σ ) − γ (Ev j,E∇E ,σ ). σt2 l( j) PE σt2 l( j) PE t2 σt2 PE t2 (V.20) The arguments that have culminated in Theorem ?? also imply that the sum (??) −4ε can be bounded by O(t2 ), for η > 4ε. The leading contribution in (??) is represented by the sum (??) of diagonal terms (with respect to G (t2)), which can now be bounded from above. It sufﬁces to show that C l( j), j,σt σt2 − 2 1 sup 2 − 2cos ∆γ (Ev j −Ev ,E∇E ,t2) e 2 ≤ lnt2 (V.21) σt2 l( j) PE η0 PE∈S t1 for some η0 > 0 that depends on ε. To see this, we note that the lower integration bound in the integral (??) contributes a factor to (??) proportional to lnt2. In Lemma ??, it is proven that 1 1 σt2 − σt2 − γ (Ev j,E∇E ,(σt ) θ ) − γ (Ev ,E∇E ,(σt ) θ ) ≤ O(|Ev j −Ev |). σt2 PE 2 σt2 l( j) PE 2 l( j) (V.22) σ σ E t1 E t2 We can estimate the difference Ev j −Evl( j) = ∇EE∗ − ∇EE∗ , which also appears Pj Pl( j) in Eηl( j), j(Ek), using condition (I 2) of Theorem ??. This yields the ε-dependent negative power of t1 in (??). 40 T. Chen, J. Frohlich,¨ A. Pizzo

V.3 Shifting the time variable for a ﬁxed cell partition and infrared cut-off

In this subsection, we prove that

∆ψ(t → t ,σ , (t1)) ≤ (ln(t ))2/tρ (V.23) 2 1 t2 G O 2 1 for some ρ > 0; see (??). This accounts for the change of the time variable, while both the cell partition and the infrared cutoff are kept ﬁxed. It can be controlled by a standard Cook argument, and methods similar to those used in the discussion of (??). For t1 ≤ s ≤ t2, we deﬁne Z s Z σt2 σt2 σt2 3 γ (Ev j,E∇E ,s) := − E∇E · ΣE (Ek)cos(Ek ·E∇E τ − |Ek|τ)d kdτ. σt2 PE PE Ev j PE 1 B S \Bσt στ 2 (V.24) Then, we estimate

σ σ Z t2 σ t2 t2 d t2 iγσ (Ev j,E∇E ,s) −iE s (t ) iH s t2 PE PE 1 ds e Wκ,σt (Ev j,s)e e ψ (V.25) 2 j,σt2 t1 ds cell by cell. To this end, we can essentially apply the same arguments that entered the treatment of the time derivative in (??), see also the remark after Theorem ??, by deﬁning an infrared tail in a similar fashion. The only modiﬁcation to be added is that, apart from two terms analogous to (??), (??), we now also have to consider

σ σ t2 t2 iγσ (Ev j,E∇E ,s) −iE s (t ) iHs σt2 t2 PE PE 1 ie (H − H )Wκ,σt (Ev j,s)e e ψ , (V.26) 2 j,σt2 which enters from the derivative in s of the operator underlined in

σ σ σ σ t2 t2 t2 t2 iγσ (Ev j,E∇E ,s) −iE s (t ) iHs −iH s iH s t2 PE PE 1 e e e Wκ,σt (Ev j,s)e e ψ . (V.27) 2 j,σt2 To control the norm of (??), we observe that

1 AE<σ · AE<σ σt t2 t2 H − H 2 = α 2 i[H,Ex] · AE<σ − α , (V.28) t2 2 where Z 3 d k n ∗ ∗ o AE< := Eε b + Eε b , (V.29) σt2 ∑ q Ek,λ Ek,λ Ek,λ Ek,λ Bσ λ=± t2 |Ek| and we note that [H,x] · AE = AE · [H,x], E <σt2 <σt2 E because of the Coulomb gauge condition. Moreover, ∗ E W (Ev j,s)i[H,Ex]Wσt (Ev j,s) = i[H,Ex] + hs(Ex) (V.30) κ,σt2 2 Infraparticle States in QED I 41

with kEhs(Ex)k < O(1) and [b ,Eh (x)] = [Eh (x), AE ] = 0. Ek,λ s E s E <σt2 Furthermore, we have

(t1) bE ψ = 0 k,λ j,σt2 forEk ∈ , and Bσt2 E (t1) E E (t1) (t1) kA<σt ψ k, kA<σt · A<σt ψ k ≤ O(σt2 kψ k). (V.31) 2 j,σt2 2 2 j,σt2 j,σt2 The estimate n o E (t1) (t1) (t1) kA<σt · [H,Ex]ψ k ≤ O σt2 k[H,Ex]ψ k + kψ k , (V.32) 2 j,σt2 j,σt2 j,σt2 holds, where

(t ) − 3ε k[H,Ex]ψ 1 k ≤ O(t 2 ), (V.33) j,σt2 1 because

(t1) σt (t1) k[H,Ex]ψ k ≤ c1k(H 2 + i)ψ k j,σt2 j,σt2 for some constant c1, and (t ) − 3ε kψ 1 k = O(t 2 ). j,σt2 1 Consequently, we obtain that σ σ t2 t2 iγσ (Ev j,E∇E ,s) −iE s (t ) σt2 t2 PE PE 1 (H − H )Wκ,σt (Ev j,s)e e ψ 2 j,σt2 n (t1) (t1) o −3/2ε ≤ O(σt2 k[H,Ex]ψ k + kψ k ) ≤ O(σt2 t ). (V.34) j,σt2 j,σt2 Following the procedure in Sect. II.2.1, B., one can also check that

σ σ Z t2 σ σ t2 t2 t2 d t2 iγσ (Ev j,E∇E ,s) −iE s (t ) iHs −iH s iH s t2 PE PE 1 e e e Wκ,σt (Ev j,s)e e ψ ds 2 j,σt2 t1 ds 3ε 1 2 − 2 ≤ O η |lnσt2 | t1 , (V.35) t1 for some η > 0 independent of ε. Similarly as in (??), we choose ε small enough η such that 4 > ε. (t1) 3ε The number of cells in the partition G is N(t1) ≈ t1 . Therefore, summing over all cells, we get

3ε 3ε − 2 1 2 − 2 O N(t1) t1 σt2t2 + O N(t1) η |lnσt2 | t1 , (V.36) t1 as an upper bound on the norm of the term in (??). −β The parameter β in the deﬁnition of σt2 = t2 can be chosen arbitrarily large, independently of ε. Hereby, we arrive at the upper bound claimed in (??). 42 T. Chen, J. Frohlich,¨ A. Pizzo

V.4 Shifting the infrared cut-off

In this section, we prove that

(t1) 2 ρ k∆ψ(t1,σt2 → σt1 ,G )k ≤ O (ln(t2)) /t1 (V.37) for some ρ > 0; see (??). The analysis of this last step is the most involved one, and will require extensive use of our previous results. The starting idea is to rewrite the last term in (??),

N(t ) σ σ 1 t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 t2 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)e e ψ , (V.38) ∑ 2 j,σt2 j=1 as

N(t ) σ σ 1 t2 t2 σt σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 2 E 2 t2 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (∇E )Wσt (∇E )e e ψ , ∑ 2 t2 PE 2 PE j,σt2 j=1 (V.39) and to group the terms appearing in (??) in such a way that, cell by cell, we consider the new dressing operator

σt σt −iE 2 t iHt1 ∗ E 2 PE 1 e Wκ,σt (Ev j,t1)W (∇E )e , (V.40) 2 σt2 PE which acts on Z σ Φ(t1) := h(PE)Φ t2 d3P, (V.41) j,σt2 (t1) PE G j

σ σ σ (t1) where Φ = Wσ (E∇E )Ψ , see (??). The key advantage is that the vector Φ PE PE PE j,σt2 inherits the Holder¨ regularity of Φσ ; see (??) in condition ( 2) of Theorem ??. PE I We will refer to (??) as an infrared-regular vector. Accordingly, (??) now reads

N(t ) σ σ 1 t2 t2 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 2 t2 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (∇E )e e Φ , (V.42) ∑ 2 t2 PE j,σt2 j=1 and we proceed as follows. A. Shifting the IR cutoff in the infrared-regular vector. First, we substitute

N(t ) σ σ 1 t2 t2 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 2 t2 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (∇E )e e Φ ∑ 2 t2 PE j,σt2 j=1

N(t ) σ σ 1 t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 2 t1 PE 1 PE 1 1 −→ e Wκ,σt (Ev j,t1)Wσ (∇E )e e Φ , ∑ 2 t2 PE j,σt1 j=1 (V.43) where σt2 is replaced by σt1 in the underlined terms. We prove that the norm difference of these two vectors is bounded by the r.h.s. of (??). The necessary ingredients are: Infraparticle States in QED I 43

1) Condition (I 1) in Theorem ??. 2) The estimate

1 σt2 σt1 2 (1−δ) 1−θ |γ (Ev j,E∇E ,t1) − γ (Ev j,E∇E ,t1)| ≤ O σ t + O (t1 σt ), σt2 PE σt1 PE t1 1 1

for t2 > t1 1, proven in Lemma ??. The parameter 0 < θ < 1 is the same as the one in (??). (t ) 3) The cell partition G 1 depends on t1 < t2. 4) The parameter β can be chosen arbitrarily large, independently of ε, so that −β the infrared cutoff σt1 = t1 can be made as small as one wishes. First of all, it is clear that the norm difference of the two vectors in (??) is bounded by the norm difference of the two underlined vectors, summed over all N(t1) cells. Using 1) and 2), one straightforwardly derives that the norm difference between the two underlined vectors in (??) is bounded from above by

1 (1− ) − 3 t 2 δ t 2 ε O( 1 σt1 1 ), (V.44)

− 3 ε 2 (t1) where the last factor, t1 , accounts for the volume of an individual cell in G , by 3). The sum over all cells in G (t1) yields a bound

1 (1− ) 1− 3 N t 2 δ t 2 ε O( ( 1)σt1 1 ), (V.45)

3ε where N(t1) ≈ t1 , by 3). Picking β sufﬁciently large, by 4), we ﬁnd that the norm −η difference of the two vectors in (??) is bounded by t1 , for some η > 0. This agrees with the bound stated in (??). B. Shifting the IR cutoff in the dressing operator. Subsequently to (??), we substitute

N(t ) σ σ 1 t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 2 t1 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (∇E )e e Φ ∑ 2 t2 PE j,σt1 j=1

N(t ) σ σ 1 t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ E 1 t1 PE 1 PE 1 1 −→ e Wκ,σt (Ev j,t1)Wσ (∇E )e e Φ (V.46) ∑ 1 t1 PE j,σt1 j=1

where σt2 → σt1 in the underlined operators. A crucial point in our argument is σ that when σ (> σ ) tends to 0, the Holder¨ continuity of Φ t1 in PE offsets the t1 t2 PE (logarithmic) divergence in t2 which arises from the dressing operator.

We subdivide the shift σt2 → σt1 in

σ σ ∗ E t2 ∗ E t1 Wκ,σt (Ev j,t1)W (∇E ) −→ Wκ,σt (Ev j,t1)W (∇E ) (V.47) 2 σt2 PE 1 σt1 PE into the following three intermediate steps, where the operators modiﬁed in each step are underlined: 44 T. Chen, J. Frohlich,¨ A. Pizzo

Step a) σ ∗ ∗ E t2 Wκ,σt (Ev j,t1)W (Ev j)Wσt (Ev j)W (∇E ) 2 σt2 2 σt2 PE σ ∗ ∗ E t2 −→ Wκ,σt (Ev j,t1)W (Ev j)Wσt (Ev j)W (∇E ), (V.48) 1 σt1 2 σt2 PE Step b) σ ∗ ∗ E t2 Wκ,σt (Ev j,t1)W (Ev j)Wσt (Ev j)W (∇E ) 1 σt1 2 σt2 PE σ ∗ ∗ E t1 −→ Wκ,σt (Ev j,t1)W (Ev j)Wσt (Ev j)W (∇E ), (V.49) 1 σt1 2 σt2 PE Step c) σ ∗ ∗ E t1 Wκ,σt (Ev j,t1)W (Ev j)Wσt (Ev j)W (∇E ) 1 σt1 2 σt2 PE σ ∗ ∗ E t1 −→ Wκ,σt (Ev j,t1)W (Ev j)Wσt (Ev j)W (∇E ). (V.50) 1 σt1 1 σt1 PE Analysis of Step a). In Step a), we analyze the difference between the vectors σ σ t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ ∗ E 2 t1 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (Ev j)Wσt (Ev j)Wσ (∇E )e e Φ 2 t2 2 t2 PE j,σt1 (V.51) and σ σ t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) iHt1 ∗ ∗ E 2 t1 PE 1 PE 1 1 e Wκ,σt (Ev j,t1)Wσ (Ev j)Wσt (Ev j)Wσ (∇E )e e Φ , 1 t1 2 t2 PE j,σt1 (V.52) for each cell in G (t1). Our goal is to prove that

k(??) − (??)k ≤ const lnt2 P(t1,t2), (V.53) where E E σt −i(|k|t1−k·Ex) ∗ E 2 P(t1,t2) := sup (e − 1)Wσt (Ev j)W (∇E )) 2 σt2 PE Ek∈Bσ t1 σ σ E t2 t1 iγσt (Ev j,∇E ,t1) −iE t1 (t1) −η ×e 2 PE e PE Φ ≤ O(t lnt2 ) (V.54) j,σt1 1 as t1 → +∞, for some η > 0 independent of ε and for β large enough. Using the identity ∗ ∗ Wκ,σt (Ev j,t1)W (Ev j) = Wκ,σt (Ev j,t1)W (Ev j) 2 σt2 1 σt1 ! iα Z Ev j · ΣEEv (Ek)sin(Ek ·Ex − |Ek|t1) × exp d3k j 2 Bσ \Bσ E t1 t2 |k|(1 −bk ·Ev j)  ∗ −i(|Ek|t −Ek·Ex)  3 Ev j ·{Eε b (e 1 − 1) − h.c.} 1 Z d k Ek,λ Ek,λ × expα 2 ∑ q , Bσ \Bσ |Ek|( − k · v ) λ t1 t2 |Ek| 1 b Ej (V.55) Infraparticle States in QED I 45 the difference between (??) and (??) is given by ! iα Z Ev j · ΣEv (Ek)sin(Ek ·Ex − |Ek|t1) iHt1 ∗ 3 Ej e Wκ,σt (Ev j,t1)W (Ev j) exp d k 1 σt1 2 Bσ \Bσ E t1 t2 |k|(1 −bk ·Ev j)   ∗ −i(|Ek|t −Ek·Ex)   3 Ev j ·{Eε b (e 1 − 1) − h.c.} 1 Z d k Ek,λ Ek,λ × expα 2 ∑ q  − I  Bσ \Bσ |Ek|( − k · v ) λ t1 t2 |Ek| 1 b Ej

σ σ t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) ∗ E 2 t1 PE 1 PE 1 1 ×Wσt (Ev j)Wσ (∇E )e e Φ (V.56) 2 t2 PE j,σt1 " ! # iα Z Ev j · ΣEv (Ek)sin(Ek ·Ex − |Ek|t1) iHt1 ∗ 3 Ej +e Wκ,σt (Ev j,t1)W (Ev j) exp d k − I 1 σt1 2 Bσ \Bσ E t1 t2 |k|(1 −bk ·Ev j) σ σ t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) ∗ E 2 t1 PE 1 PE 1 1 ×Wσt (Ev j)Wσ (∇E )e e Φ , (V.57) 2 t2 PE j,σt1 where I is the identity operator in H . The norm of the vector (??) equals

  ∗ −i(|Ek|t −Ek·Ex)   Z 3 Ev j ·{Eε b (e 1 − I ) − h.c.} 1 d k Ek,λ Ek,λ expα 2 ∑ q  − I  Bσ \Bσ |Ek|(1 − k · v ) λ t1 t2 |Ek| b Ej

σ σ t1 t1 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) ∗ E 2 t1 PE 1 PE 1 1 ×Wσt (Ev j)Wσ (∇E )e e Φ j,σ . (V.58) 2 t2 PE t1

We now observe that • forEk ∈ , Bσt1 σ ∗ E t2 b Wσt (Ev j)W (∇E ) (V.59) Ek,λ 2 σt2 PE σ ∗ E t2 = Wσt (Ev j)W (∇E )b (V.60) 2 σt2 PE Ek,λ σ ∗ E t2 +Wσt (Ev j)W (∇E ) f (Ev j,PE), (V.61) 2 σt2 PE Ek,λ where Z 3 E 2 d k| fEk,λ (Ev j,P)| ≤ O(|lnσt2 |) (V.62) Bσ \Bσ t1 t2

uniformly in Ev j, and in PE ∈ S , and where j enumerates the cells. • forEk ∈ , Bσt1

σt σt i (v ,E∇E 1 ,t ) −iE 1 t (t ) γσt Ej E 1 E 1 1 bE e 2 P e P Φ = 0, (V.63) k,λ j,σt1

because of the infrared properties of Φ(t1) . j,σt1 From the Schwarz inequality, we therefore get

(??) ≤ c|lnσt2 |P(t1,t2), (V.64) 46 T. Chen, J. Frohlich,¨ A. Pizzo for some ﬁnite constant c as claimed in (??), where

E E σt −i(|k|t1−k·Ex) ∗ E 2 P(t1,t2) = sup (e − 1)Wσt (Ev j)W (∇E ) 2 σt2 PE Ek∈Bσ t1 σ σ t2 t2 iγσ (Ev j,E∇E ,t ) −iE t (t ) ×e t2 PE 1 e PE 1 Φ 1 , (V.65) j,σt1 as deﬁned in (??). To estimate P(t1,t2), we regroup the terms inside the norm into

σ σ t2 t2 E E σt iγσ (Ev j,E∇E ,t ) −iE t (t ) −i(|k|t1−k·Ex) ∗ E 2 t2 PE 1 PE 1 1 (e − 1)Wσt (Ev j)Wσ (∇E )e e Φ 2 t2 PE j,σt1 σ σ t2 t2 σt E E iγσ (Ev j,E∇E ,t ) −iE t (t ) ∗ E 2 −i(|k|t1−k·Ex) t2 PE 1 PE 1 1 = Wσt (Ev j)Wσ (∇E )(e − I )e e Φ 2 t2 PE−Ek j,σt1 (V.66) σ σ t2 t2 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) ∗ E 2 t2 PE 1 PE 1 1 + Wσt (Ev j)Wσ (∇E )e e Φ (V.67) 2 t2 PE−Ek j,σt1 σ σ t2 t2 σt iγσ (Ev j,E∇E ,t ) −iE t (t ) ∗ E 2 t2 PE 1 PE 1 1 − Wσt (Ev j)Wσ (∇E )e e Φ . (V.68) 2 t2 PE j,σt1 We next prove that

ρ k(??)k , k(??) − (??)k ≤ O((σt1 ) t1 lnt2 ) (V.69) for some ρ > 0 independent of ε. To this end, we use: i) The Holder¨ regularity of Φσ and E∇Eσ described under condition ( 2) in PE PE I Theorem ??. ii) The regularity of the phase function

σt2 γ (Ev j,E∇E ,t1) (V.70) σt2 PE

with respect to PE ∈ supph ⊂ S expressed in the following estimate, which is similar to (??) in Lemma ??: ForEk ∈ and t large enough, Bσt1 1

1 00 σt2 σt2 (1−δ ) (1−θ) γσ (Ev j,E∇E ,t1) − γσ (Ev j,E∇E ,t1) ≤ O(|Ek| 4 t ), (V.71) t2 PE t2 PE−Ek 1

where 0 < θ(< 1) can be chosen arbitrarily close to 1. iii) The estimate

E σ 1σ,Λ (k) kbE Ψ k ≤ C (V.72) k,λ PE |Ek|3/2

from (I 5) in Theorem ?? for PE ∈ S , which implies

1/2 Z ! kN1/2 σ k = d3kkb σ k2 ≤ C | |1/2. f ΨPE ∑ Ek,λ ΨPE lnσ (V.73) λ Infraparticle States in QED I 47

Likewise,

!1/2 Z 2 kN1/2 Φσ k = d3k b + (|Ek|−3/2) Ψ σ f PE ∑ Ek,λ O PE λ BΛ \Bσ ≤ C | lnσ |1/2, (V.74)

σ which controls the expected photon number in the states {Φ t1 }. As a side PE remark, we note that the true size is in fact O(1), uniformly in σ, but the logarithmically divergent bound here is sufﬁcient for our purposes. (t ) iv) The cell decomposition G 1 is determined by t1 < t2. Moreover, since β(> −β 1) can be chosen arbitrarily large and independent of ε, σt1 = t1 can be made as small as desired. We ﬁrst prove the bound on k(??)k stated in (??). To this end, we use

σ σ t1 t1 −i(|Ek|t −Ek·x) iγσ (Ev j,E∇E ,t ) −iE t (t ) e 1 E − I e t2 PE 1 e PE 1 Φ 1 (V.75) j,σt1 σ σ t1 t1 iγσ (Ev j,E∇E ,t1) −iE t1 i Ek t Ek (t ) = e t2 PE−Ek e PE−Ek (e− (| | 1− ·Ex) − I )Φ 1 (V.76) j,σt1 σ σ t1 t1 iγσ (Ev j,E∇E ,t ) −iE t (t ) + e t2 PE−Ek 1 e PE−Ek 1 Φ 1 (V.77) j,σt1 σ σ t1 t1 iγσ (Ev j,E∇E ,t ) −iE t (t ) − e t2 PE 1 e PE 1 Φ 1 . (V.78) j,σt1

σ The Holder¨ regularity of Φ t1 from i) yields PE

( 1 − 0) − 3ε t 4 δ t 2 k(??)k ≤ O( 1 σt1 1 ), (V.79) where δ 0 can be chosen arbitrarily small, and independently of ε. The derivation of a similar estimate is given in the proof of Theorem ?? in the Appendix, starting σ σ from (??), to which we refer for details. The Holder¨ continuity of E t1 and E∇E t1 , PE PE again from i), combined with ii), with θ sufﬁciently close to 1, implies that, with Ek ∈ , Bσt1

1 − 3ε t 5 t 2 k(??) − (??)k ≤ O( 1 σt1 1 ), (V.80) as desired. To prove the bound on k(??) − (??)k asserted in (??), we write

σt σt σt σt σt W ∗ (E∇E 2 ) −W ∗ (E∇E 2 ) = W ∗ (E∇E 2 )(W ∗ (E∇E 2 ; E∇E 2 ) − I ), σt2 PE−Ek σt2 PE σt2 PE σt2 PE−Ek PE (V.81) where σ σ σ σ ∗ E t2 E t2 E t2 ∗ E t2 W (∇E ; ∇E ) := Wσt (∇E )W (∇E ), σt2 PE−Ek PE 2 PE σt2 PE−Ek 48 T. Chen, J. Frohlich,¨ A. Pizzo and apply the Schwarz inequality in the form

σt σt (W ∗ (E∇E 2 ; E∇E 2 ) − I )Φe σt2 PE+Ek PE 1/2 3 ! Z d q σ σ ≤ C sup E∇E t2 −E∇E t2 kN1/2Φ k, (V.82) 3 PE−Ek PE f e Bσ |Eq| t2 PE∈supph,Ek∈Bσ t1

σ σ t1 t1 iγσ (Ev j,E∇E ,t ) −iE t (t ) where in our case, Φe ≡ e t2 PE 1 e PE 1 Φ 1 . We have j,σt1 σ kN1/2 Φ t1 k ≤ c|lnσ |1/2 ≤ c0 (lnt )1/2, (V.83) f PE t1 1 as a consequence of iii). Due to i),

σ σ 1 −δ 00 sup E∇E t2 − E∇E t2 ≤ (σ 4 ), (V.84) PE−Ek PE O t1 PE∈supph,Ek∈Bσ t1 where δ 00 > 0 is arbitrarily small, and independent of ε (see (??)). Therefore,

ρ0 sup k(??) − (??)k ≤ O((lnt2)(σt1 ) ) Ek∈Bσ t1

0 for some ρ > 0 which does not depend on ε (recalling that t1 < t2). We may now return to (??). From iv), and the fact that the number of cells is 3ε N(t1) ≈ t1 , summation over all cells yields

N(t1) ln(t2) ∑ k(??) − (??)k ≤ O( ρ ) (V.85) j=1 t1 for some ρ > 0, provided that β is sufﬁciently large. This agrees with (??). N(t1) The sum ∑ j=1 k(??)k can be treated in a similar way. Analysis of Step b). To show that the norm difference of the two vectors corresponding to the change (??) in (??) is bounded by the r.h.s. of (??), we argue similarly as for Step a), and we shall not reiterate the details. One again uses properties i) – iv) as in Step a). Analysis of Step c). Finally, we prove that the difference of the vectors corresponding to (??) satisﬁes

N(t ) σ σ 1 t1 t1 2 iHt h σt ∗ σt σt i iγσ (Ev j,E∇E ,t ) −iE t (t ) 1 1 1 E 1 t1 PE 1 PE 1 1 e Wκ,σt (Ev j,t1) W|σt (Ev j)W |σt (∇E )−I e e ψ ∑ 1 2 2 PE j,σt1 j=1 2 ρ ≤ O (ln(t2)) /t1 (V.86) for some ρ > 0, where we deﬁne

σt1 ∗ W|σ (Ev j) := W (Ev j)Wσt (Ev j), (V.87) t2 σt1 2 Infraparticle States in QED I 49 and likewise,

σ σ σ σ ∗ t1 E t1 ∗ E t1 E t1 W |σ (∇E ) := W (∇E )Wσt (∇E ). (V.88) t2 PE σt2 PE 1 PE We separately discuss the diagonal and off-diagonal contributions to (??) from the sum over cells in G (t1). • The diagonal terms in (??). To bound the diagonal terms in (??), we use that, E σt with Ev j ≡ ∇E 1 |E E∗ , P=Pj

σt1 σt1 σt1 ∗ σt1 σt1 W| (Ev j ; E∇E ) := W| (Ev j)W | (E∇E ) (V.89) σt2 PE σt2 σt2 PE allows for an estimate similar to (??), where we now use that

σ −ε( 1 −δ 00) sup |E∇E t1 − v | ≤ (t 4 ). (V.90) PE Ej O 1 E (t1) P∈G j

The latter follows from the Holder¨ regularity of E∇Eσ , due to condition ( 2) in PE I Theorem ??; see (??). Moreover, we use (??) to bound the expected photon num- σ ber in the states {Ψ t1 }. PE Hereby we ﬁnd that the sum of diagonal terms can be bounded by

1 00 (t1) 2 −ε( 4 −δ ) −ρ O(N(t1)kψ k (lnt2)t ) ≤ O(t lnt2 ) (V.91) j,σt1 1 1

3ε (t1) 2 −3ε for some ρ > 0, using N(t1) = O(t ), and kψ k = O(t ). 1 j,σt1 1 • The off-diagonal terms in (??). Next, we bound the off-diagonal terms in (??), corresponding to the inner product of vectors supported on cells j 6= l of the par- (t1) 1 tition G . Those are similar to the off-diagonal terms Mbl, j(t,s) in (??) that were discussed in detail previously. Correspondingly, we can apply the methods developed in Sect. ??, up to some modiﬁcations which we explain now. Our goal is to prove the asymptotic orthogonality of the off-diagonal terms in (??). We ﬁrst of all prove the auxiliary result

σ σ σt ∗ t1 E t1 −iH 1 s (t1) lim kEaσt (Eηl, j)(s)W |σt (∇E )e ψ j, k = 0. (V.92) s→+∞ 1 2 PE σt1

To this end, we compare

∗ σt1 σt1 W | (E∇E )1 (t ) (PE) (V.93) σt2 E 1 P G j

(t1) (where 1 (t1) is the characteristic function of the cell G j ) to its discretization: G j

1. We pick t¯ large enough such that G (t¯) is a sub-partition of G (t); in particular, (t) M (t¯) N(t¯) G j = ∑m( j)=1 Gm( j), where M = N(t) . 50 T. Chen, J. Frohlich,¨ A. Pizzo

σ E t1 E∗ 2. Furthermore, deﬁning Eum( j) := ∇EE∗ , where Pm( j) is the center of the cell Pm( j) (t¯) E (t¯) Gm( j), we have, for P ∈ Gm( j),

1 00 ε ( 4 −δ ) σt 1 |Eu −E∇E 1 | ≤ C , (V.94) m( j) PE t¯

where C is uniform in t1. 3. We deﬁne

M σt1 ∗ σt1 σ (M) := W |σ (Eum( j))1 (t¯) (PE) (V.95) W t2 ∑ t2 G m( j)=1 m( j)

and rewrite the vector

σ σ σt ∗ t1 E t1 −iH 1 s (t1) Eaσt (Eηl, j)(s)W |σt (∇E )e ψ (V.96) 1 2 PE j,σt1 in (??) as

Z h σ σ σ i d3ke−iEk·Ex W ∗| t1 (E∇E t1 ) − t1 (M) ∑ σt2 PE Wσt2 Bκ \Bσ λ t1 σt E −iE 1 s (t ) ∗ i|k|s E 1 ×Eηl, j(Ek) ·Eε bE e e P ψ (V.97) Ek,λ k,λ j,σt1 M Z ∗ σt1 3 ∗ i|Ek|s + W | (Eu ) d kEηl, j(Ek) ·Eε a e ∑ σt2 m( j) ∑ Ek,λ Ek,λ Bκ \Bσ m( j)=1 λ t1 σt −iE 1 s (t¯) ×e PE ψ . (V.98) m( j),σt1

We now observe that, at ﬁxed t1,t2,(??) and the bound (??) imply that the vector in (??) converges to the zero vector as t¯ → +∞, uniformly in s. Moreover, the norm of the vector in (??) tends to zero, as s → +∞, at ﬁxed t¯. This proves (??). The main difference between the vector corresponding to the jth cell in (??) and the similar expression in (??) that is differentiated in s is the term proportional to the operator

σt1 ∗ σt1 σt1 W| (Ev j)W | (E∇E ), (V.99) σt2 σt2 PE which is absent in (??). To control it, we ﬁrst note that the Hamiltonian

Z ⊕ σ σt 3 H t1 := H 1 d P, (V.100) b bPE where

f 2 E E 1/2Eσt1 P − P>σt + α A σt1 1 f Hb := + H>σ (V.101) PE 2 t1 Infraparticle States in QED I 51 with Z f ∗ 3 PE>σ := Ekb bE d k, (V.102) t1 3 Ek,λ k,λ \Bσ R t1 and Z f ∗ 3 H>σ := |Ek|b bE d k, (V.103) t1 3 Ek,λ k,λ \Bσ R t1 satisﬁes σ σ σ σ H t1 Ψ t1 = E t1 Ψ t1 , (V.104) bPE PE PE PE and

σt1 ∗ σt1 σt1 σt [W| (Ev j)W | (E∇E ), H 1 ] = 0. (V.105) σt2 σt2 PE b Using (??), the vector in (??) corresponding to the jth cell can be replaced by

σ σ t1 σ t1 iγσ (Ev j,∇E ,s) t1 σt σt (t) iHb s t1 PE −iHb s 1 E 1 e Wκ,σt (Ev j,s)e e W (Ev j;∇E )−I ψ j,σt , for s=t1, 1 σt2 PE 1 (V.106) where we recall that

σt1 σt1 σt1 ∗ σt1 σt1 W| (Ev j ; E∇E ) = W| (Ev j)W | (E∇E ). σt2 PE σt2 σt2 PE

1 Similarly to our strategy in Sect. ??, we control Mbl, j(t,s); see (??). The deriva- σt σt1 tive in s of the term proportional to W 1 (Ev j ; E∇E ) in the jth cell vector has the σt2 PE form

σ σ σ t1 t1 d t1 iγσ (Ev j,E∇E ,s) −iE s σt σt (t ) iHb s t1 PE PE 1 E 1 1 e Wκ,σt (Ev j,s)e e W|σt (Ev j ; ∇E )ψ j,σ ds 1 2 PE t1 σ Z iH t1 s σ 3 = ie b (v ,s)α i[H t1 ,x] · ΣE (Ek)cos(Ek · x − |Ek|s)d k Wκ,σt1 Ej b E Ev j E Bκ \Bσ t1 σ t1 σt −iE s iγσt (Ev j,∇E ,s) σt1 σt1 (t) ×e PE e 1 PE W| (Ev j ; E∇E )ψ σt2 PE j,σt σt Z +ieiHb 1 s (v ,s)α2 ΣE (Ek) cos(Ek · x − |Ek|s)d3k Wκ,σt1 Ej Ev j E Bκ \Bσ t1 Z E 3 · ΣEv j (Eq)cos(Eq ·Ex − |Eq|s)d q Bκ \Bσ t1 σ σt t1 iγσ (Ev j,∇E ,s) −iE s σt σt (t) t1 PE PE 1 E 1 ×e e W|σt (Ev j ; ∇E )ψ 2 PE j,σt1 σt1 σt dγσ (Ev j,∇E ,s) iHb 1 s t1 PE +ie Wκ,σ (Ev j,s) t1 ds σ σ t1 t1 iγσ (Ev j,E∇E ,s) −iE s σt σt (t ) t1 PE PE 1 E 1 1 ×e e W|σt (Ev j ; ∇E )ψ . (V.107) 2 PE j,σt1 52 T. Chen, J. Frohlich,¨ A. Pizzo

Due to the similarity of this expression with (??)–(??), we can essentially adopt σ the analysis presented in Sect. ??. The only difference here is the operator Hb t1 σ instead of H t1 , and the additional term involving the commutator Ex ∗ σt1 σt1 χh , W |σ E∇E (V.108) s t2 PE applied to the one-particle state

σ σ t1 t1 iγσ (Ev j,E∇E ,s) −iE s (t ) e t1 PE e PE ψ 1 . (V.109) j,σt1

1 However, the latter tends to zero as s → +∞, at a rate O( sη ), for some ε-independent η > 0. This follows from the Holder¨ regularity of E∇Eσ (condition ( 2) in The- PE I orem ??), and (??). Similarly, we treat the commutator (??) with the infrared tail Ex σt1 σt1 (??) in place of χh( s ) (and with Hb replacing H ). It is then straightforward to see that we arrive at (??). ut

VI Scattering Subspaces and Asymptotic Observables

This section is dedicated to the following key constructions in the scattering theory for an infraparticle with the quantized electromagnetic field: i) We define scattering subspaces H out/in which are invariant under space- out/in time translations, built from vectors {Ψh,κ }. ˚ out/in To this end, we first define a subspace, Hκ , depending on the choice of a threshold frequency κ with the following purpose: Apart from photons with energy smaller than κ, this subspace contains states describing only a freely moving (asymptotic) electron. ˚ out/in Adding asymptotic photons to the states in Hκ , we define spaces of scattering states of the system, where the asymptotic electron velocity is E E 1 restricted to the region {∇EPE : |P| < 3 }. ˚ out/in We note that the choice of Hκ is not unique, except for the behavior of the dressing photon cloud in the infrared limit. It is useful because – in the construction of the spaces of scattering states, we can separate ˚ out/in “hard photons” from the photon cloud present in the states in Hκ , which is not completely removable – each state in the scattering spaces contains an infinite number of asymptotic photons. – from the physical point of view, every experimental setup is limited by a threshold energy κ below which photons cannot be measured. out/in out/in ii) The construction of asymptotic algebras of observables, Aph and Ael , related to the electromagnetic field and to the electron, respectively. The asymptotic algebras are out/in – the Weyl algebra, Aph , associated to the asymptotic electromagnetic field; Infraparticle States in QED I 53

out/in – the algebra Ael generated by smooth functions of compact support of the asymptotic velocity of the electron. out/in out/in The two algebras Aph and Ael commute. This is the mathematical counterpart of the asymptotic decoupling between the photons and the electron. This decoupling is, however, far from trivial: In fact, in contrast to a theory with a mass gap or a theory where the interaction with the soft modes of the field is turned off, the system is characterized by the emission of soft photons for arbitrarily long times. In this respect, the asymptotic convergence of the electron velocity is a new conceptual result, obtained from the solution of the infraparticle problem in a concrete model, here non-relativistic QED. Furthermore, the emission of soft photons for arbitrarily long times is reflected in the representation of the asymptotic electromagnetic algebra, which is non-Fock but only locally Fock (see Sect. ??). More precisely, the representation can be decomposed on the spectrum of the asymptotic velocity of the electron; for different values of the asymptotic velocity, the representations turn out to be E inequivalent. Only for ∇EPE = 0, the representation is Fock, otherwise they are coherent non-Fock. The coherent photon cloud, labeled by the asymptotic velocity, is the well known Bloch-Nordsieck cloud. All the results and definition clearly hold for both the out and the in-states. We shall restrict ourselves to the discussion of out-states.

VI.1 Scattering subspaces and “One-particle” subspaces with counter threshold κ

In Sect. ??, we have constructed a scattering state with electron wave function h, and a dressing cloud exhibiting the correct behavior in the limit Ek → 0, with maximal photon frequency κ. To construct a space which is invariant under space-time translations, we may either focus on the vectors

−iEa·PE −iHτ out e e ψh,κ , (VI.1) or on the vectors obtained from

N(t) E σt σt iHt τ,Ea iγσt (Ev j,∇E ,t) −iE t (t) s − lim e Wκ,σ (Ev j,t)e PE e PE ψ (τ,Ea), (VI.2) t→+∞ ∑ t j,σt j=1 where

 ∗ −i|Ek|(t+τ) −iEk·Ea  Z 3 Ev j ·{EεE a e e − h.c.} τ,Ea 1 d k k,λ Ek,λ Wκ,σ (Ev j,t) := expα 2 , t ∑ q E λ Bκ \Bσt |Ek| |k|(1 −bk ·Ev j) (VI.3) and

Z σ (t) −ia·PE −iE t τ σ 3 ψ (τ,a) := e E e PE h(PE)ψ t d P. (VI.4) j,σt E (t) PE G j 54 T. Chen, J. Frohlich,¨ A. Pizzo

Using Theorem ??, one straightforwardly ﬁnds that

−iEa·PE −iHτ out e e ψh,κ (VI.5) N(t) E σt σt −iEa·PE −iHτ iHt iγσt (Ev j,∇E ,t) −iE t (t) = s − lim e e e Wκ,σ (Ev j,t)e PE e PE ψ (VI.6) t→+∞ ∑ t j,σt j=1 N(t+τ) E σt+τ σt+τ iHt τ,Ea iγσt+τ (Ev j,∇E ,t+τ) −iE t (t+τ) = s − lim e Wκ,σ (Ev j,t)e PE e PE ψ (τ,Ea). t→+∞ ∑ t+τ j,σt+τ j=1 (VI.7)

The two limits (??) and (??) coincide; this follows straightforwardly from the line of analysis presented in the previous section. Therefore, we can deﬁne the “one-particle” space corresponding to the frequency threshold κ as

n o ˚ out/in _ out/in E 1 3 Hκ := ψh,κ (τ,Ea) : h(P) ∈ C0(S \Brα ), τ ∈ R, Ea ∈ R . (VI.8)

˚ out/in By construction, Hκ is invariant under space-time translations. Infraparticle States in QED I 55

General scattering states of the system can contain an arbitrarily large number of “hard” photons, i.e., photons with an energy above a frequency threshold, say ˚ out/in for instance κ. One can construct such states based on Hκ according to the following procedure. We consider positive energy solutions of the form Z 3 d k −i|k|t+iEk·Ey Ft (Ey) := Fb(Ek)e (VI.9) 2(2π)3p|k| of the free wave equation 2 ∂ Ft (Ey) E∇ ·E∇ Ft (Ey) − = 0, (VI.10) Ey Ey ∂t2 E ∞ 3 which exhibit fast decay in |Ey| for arbitrary ﬁxed t, and where Fb(k) ∈C0 (R \{0}). We then construct vector-valued test functions Z 3 d k ∗ λ −i|k|t+iEk·Ey FEt (Ey) := Eε F (Ek)e (VI.11) ∑ 3p Ek,λ b λ=± 2(2π) |k| satisfying the wave equation (??), with

FEb(Ek) := Eε ∗ Fλ (Ek) ∈ C∞( 3\{0}; 3). (VI.12) ∑ Ek,λ b 0 R C λ We set AE(t,Ey) := eiHt AE(Ey)e−iHt ; (VI.13) here AE(Ey) is the expression in (??) with Λ = ∞. An asymptotic vector potential is constructed starting from LSZ (t → ±∞) limits of interpolating field operators, Z ! ∂FEt (Ey) ∂AE(t,Ey) 3 AE[FEt ,t] := i AE(t,Ey) · − · FEt (Ey) d y, (VI.14) ∂t ∂t with FEt as in (??) for the negative-energy component, and with −FEt for the positive- energy component. We define out/in ψ := s − lim ψh,FE (t), (VI.15) h,FE t→+/−∞ where i AE[FEt ,t]−AE[FEt ,t] ψh,FE (t) := e ψh,κ (t). (VI.16) out ˚ out/in Here, ψh,κ (t) approximates a vector ψh,κ in Hκ (we temporarily drop the dependence on (τ,Ea) in our notation). The existence of the limit in (??) is a straightforward consequence of standard decay estimates for oscillating integrals under the assumption in (??), combined with the propagation estimate (??). Finally, we can define the scattering subspaces as

n_ out/in o out/in := ψ : h(PE) ∈ C1( \ ), FEb ∈ C∞( 3\0; 3) . (VI.17) H h,FE 0 S Brα 0 R C 56 T. Chen, J. Frohlich,¨ A. Pizzo

VI.2 Asymptotic algebras and Bloch-Nordsieck coherent factor

We now state some theorems concerning the construction of the asymptotic algebras. The proofs can be easily derived using the arguments developed in Sects. ?? and ??; for further details we refer to ?.

∞ 3 iHt Ex −iHt Theorem VI1 The functions f ∈ C0 (R ), of the variable e t e , have strong limits for t → ∞ in H out/in, namely: Ex out/in out/in s − lim eiHt f e−iHt ψ =: ψ , (VI.18) h,FE h f ,FE t→+/−∞ t E∇E where f (PE) := lim f (E∇Eσ ). E∇E σ→0 PE The proof is obtained from an adaptation of the proof of Theorem ?? in the Ap- pendix. For the radiation ﬁeld, we have the following result. Theorem VI2 The LSZ Weyl operators i AE[GEt ,t]−AE[GEt ,t] λ 2 3 −1 3 e : Gb (Ek) ∈ L (R ,(1 + |Ek| )d k), λ = ± (VI.19) have strong limits in H out/in: i AE[GE ,t]−AE[GE ,t] W out/in(GE) := s − lim e t t . (VI.20) t→+/−∞ The limiting operators are unitary, and have the following properties: i)

E E0 out/in out/in 0 out/in 0 − ρ(G,G ) W (GE)W (GE ) = W (GE + GE )e 2 , (VI.21) where ! Z λ ρ(GE,GE0) = 2iIm ∑ Gbλ (Ek)Gb0 (Ek)d3k . (VI.22) λ

out/in ii) The mapping R 3 s −→ W (sGE) deﬁnes a strongly continuous, one parameter group of unitary operators. iii)

iHτ out/in −iHτ out/in e W (GE)e = W (GE−τ ), (VI.23)

where GE−τ is a freely evolved, vector-valued test function in the time −τ. Next, we deﬁne out/in • Ael as the norm closure of the (abelian) *algebra generated by the limits in (??). Infraparticle States in QED I 57

out/in • Aph as the norm closure of the *algebra generated by the unitary operators in (??). out/in From (??) and (??), we conclude that Aph is the Weyl algebra associated to a free radiation field. Moreover, from straightforward approximation arguments out/in out/in applied to the generators, we can prove that the two algebras, Ael and Aph , commute. Moreover, we can next establish key properties of the representation Π of the out/in algebras Aph for the concrete model at hand that confirm structural features derived in ? under general assumptions. out/in To study the infrared features of the representation of Aph , it suffices to analyze the expectation of the generators {W out (GE)} of the algebra with respect out to arbitrary states of the form ψh,κ , out out E out ψh,κ , W (G)ψh,κ (VI.24)

N(t) σ σ D iγ (Ev ,E∇E t ,t) −iE t t (t) = lim e σt l PE e PE ψ , (VI.25) t→+∞ ∑ l,σt j=1,l=1

σt σt ∗ i AE[GEt ,0]−AE[GEt ,0] iγ (Ev j,E∇E ,t) −iE t (t) E (v ,t)e (v ,t)e σt PE e PE . Wκ,σt El Wκ,σt Ej ψ j,σt In the step passing from (??) to (??), we use Theorem ??. One infers from the arguments developed in Sect. ?? that the sum of the off-diagonal terms, l 6= j, vanishes in the limit. Therefore,

N(t) σ σ D iγ (Ev ,E∇E t ,t) −iE t t (t) (??) = lim e σt j PE e PE ψ , t→+∞ ∑ j,σt j=1

σt σt i AE[GEt ,0]−AE[GEt ,0] ρv (GE) iγ (Ev ,E∇E ,t) −iE t (t) E e e Ej e σt j PE e PE , (VI.26) ψ j,σt where Eu ·Eε ∗ ! 1 Z Ek,λ E 2 λ E 3 ρEu(G) := 2iRe α ∑ Gb (k) 3 d k . (VI.27) λ Bκ |Ek| 2 (1 −Eu ·bk) After solving an ODE analogous to (??), we ﬁnd that the diagonal terms yield

Z CE ρ (GE) out out out − G E∇E E 2 PE 2 3 ψh,κ , W (G)ψh,κ = e e |h(PE)| d P, (VI.28) where Z bE E 2 3 CGE := |G(k)| d k. (VI.29)

E σt E σt E Here, we also use that Ev j ≡ ∇EE∗ , combined with the convergence ∇EE → ∇EPE Pj P (as t → ∞ and PE ∈ S ). 58 T. Chen, J. Frohlich,¨ A. Pizzo

out/in Now, we can reproduce the following results in ?: The representation Π(Aph ) out/in out/in is given by a direct integral on the spectrum of the operator Evas in H , de- ﬁned by out/in iHt Ex −iHt f (Evas ) := s − lim e f e (VI.30) t→+/−∞ t

∞ 3 for any f ∈ C0 (R ), of mutually inequivalent, irreducible representations. These out/in representations are coherent non-Fock for values Evas 6= 0. The coherent factors, out/in labeled by Evas , are

out/in out/in ∗ Evas ·Eε 1 Evas ·EεEk,λ 1 Ek,λ 2 2 , α 3 out/in and α 3 out/in (VI.31) |Ek| 2 (1 −Evas ·bk) |Ek| 2 (1 −Evas ·bk) for the annihilation and the creation part, aout/in and aout/in∗, respectively. Ek,λ Ek,λ out/in The representation Π(Aph ) is locally Fock in momentum space. This property is equivalent to the following one: Eκ ∞ 3 3 For any κ > 0, and Gc ∈ C0 (R \Bκ ; C ), the operator

κ AE[−GEt ,t] (VI.32)

out annihilates vectors of the type ψh,κ in the limit t → +∞, i.e.,

κ out lim AE[−GEt ,t]ψ = 0. (VI.33) t→+∞ h,κ To prove this, we ﬁrst consider Theorem ??, then

κ out κ lim AE[−GEt ,t]ψ = lim AE[−GEt ,t]ψh,κ (t). (VI.34) t→+∞ h,κ t→+∞ Next, we rewrite the vector

N(t) σt σt iHt iγ (Ev j,E∇E ,t) −iE t (t) AE[−GEκ ,t] (t) = e AE[−GEκ ,0] (v ,t)e σt PE e PE t ψh,κ t ∑ Wκ,σt Ej ψ j,σt j=1 (VI.35) as

N(t) Z +∞ σt σt d iHs iγ (Ev j,E∇E ,s) −iE s (t) − {e AE[−GEκ ,0] (v ,s)e σt PE e PE }ds s ∑ Wκ,σt Ej ψ j,σt t ds j=1 (VI.36) N(t) σt σt iHs iγ (Ev j,E∇E ,s) −iE s (t) E Eκ σt PE PE + lim e Wκ,σt (Ev j,s)A[−Gs ,0]e e ψ . (VI.37) s→+∞ ∑ j,σt j=1 Infraparticle States in QED I 59

The integral in (??), and the limit in (??) exist. To see this, it is enough to follow the procedure in Sect. ??, taking into account that the operator

1 κ σt AE[−GEs ,0] [H ,Ex] (VI.38) (Hσt + i) is bounded, uniformly in t and s. The limit (??) vanishes at fixed t because of condition (I 4) in Theorem ??. Therefore we finally conclude that the limit (??) vanishes. Lienard-Wiechert´ fields generated by the charge. Now we briefly explain how to obtain the result stated in (??). The assertion is obvious for the longitudinal degrees of freedom; see the definition of Fµν in (??). For the transverse degrees of freedom, we argue as follows. Similarly to the treatment of (??), we arrive at a sum over the diagonal terms,

Z out/in iHt 3 tr ˜ −iHt out/in lim ψ , e d y F (0,Ey)δΛ (Ey −Ex − dE)e ψ t→±∞ h,FE µν h,FE N(t) Z σt 3 tr ˜ σt = lim ψ , d y F (0,Ey)δΛ (Ey −Ex − dE)ψ . t→±∞ ∑ j,σt µν j,σt j=1

Then, one uses the pull-through formula as in Lemma 6.1 in ?, and Proposition 5.1 in ? which identiﬁes the infrared coherent factor by showing that

E D E 1 1σ,Λ (k) 1 1/2 −1 Ψ σ , b Ψ σ + α 2 Eε ·E∇Eσ ≤ α C |Ek| PE Ek,λ PE 1 σ Ek,λ PE E 2 |Ek| −Ek ·E∇E |k| PE (VI.39) forEk → 0. These ingredients imply that ( N(t) Z 2 σt 3 tr ˜ σt |dE| lim ψ , d y F (0,Ey)δΛ (Ey −Ex − dE)ψ t→±∞ ∑ j,σt µν j,σt j=1 N(t) tr ) Z D E E∇Eσt − |h(PE)|2 ψσt , ψσt F PE (0,dE)d3P ≤ (|dE|−1/2), ∑ (t) PE PE µν O j=1 G j (VI.40) which vanishes in the limit |dE| → ∞, as asserted in (??).

Appendix A

In the Appendix, we present detailed proofs of auxiliary results used in Sect. ??.

Lemma A1 The following estimates hold for PE ∈ S : 60 T. Chen, J. Frohlich,¨ A. Pizzo

(i) For t2 > t1 1,

1 1 σt2 − σt2 − |γ (Ev j,E∇E ,(σt ) θ ) − γ (Ev ,E∇E ,(σt ) θ )| ≤ O(|Ev j −Ev |), σt2 PE 2 σt2 l( j) PE 2 l( j) (A.1)

σ σ E t1 E t2 where Ev j ≡ ∇EE∗ and Ev`( j) ≡ ∇EE∗ . Pj P`( j) (ii) For t2 > t1 1,

1 σt2 σt1 (1−δ) 1−θ |γ (Ev j,E∇E ,t1) − γ (Ev j,E∇E ,t1)| ≤ O [(σt ) 2 t +t1 σt ] . σt2 PE σt1 PE 1 1 1 (A.2)

(iii) For s,t 1 and Eq ∈ {Eq : |Eq| < s(1−θ)},

σ σ − θ (1− 00) (1− ) |γ (v ,E∇E t ,s) − γ (v ,E∇E t ,s)| ≤ (s 4 δ s θ ), (A.3) σt Ej PE σt Ej E Eq O P+ s

whenever γ (v ,E∇Eσt ,s) is deﬁned. σt Ej PE Proof The proofs only require the deﬁnition of the phase factor, and some elementary integral estimates, using conditions (I 1) and (I 2) in Theorem ??. ut Lemma A2 For s ≥ t 1, the estimates Z |ln σ | sup l (Ek)cos(Ek · x − |Ek|s)d3k ≤ ( t ), (A.4) ΣEv j E O 3 \ s Ex∈R Bκ Bσt

Z Ex tθ sup Σ l (Ek)cos(Ek · x − |Ek|s)d3k χ ( ) ≤ ( ), (A.5) Ev j E h O 2 x∈ 3 Bκ \B S s s E R σt hold, where

l l0 l k k l0 1 Σ (Ek) := 2 (δl,l0 − )v , (A.6) Ev j ∑ 2 j 2 l0 |Ek| |Ek| (1 −bk ·Ev j)

−β S −θ 1 and where σt := t , στ := τ , with β > 1, 0 < θ < 1. Moreover, χh(Ey) = 0 for |Ey| ≤ 2 νmin 1+νmax and |Ey| ≥ 2 with 0 < νmin < νmax < 1 (see (??)). Proof To prove the estimate (??), we consider the variable Ex ﬁrst in the set

3 {Ex ∈ R : |Ex| < (1 − ρ)s, 0 < ρ < 1}. E E We denote by θEk the angle between Ex and k. Integration with respect to |k| yields Z l (Ek)cos(Ek · x − |Ek|s)d3k (A.7) ΣEv j E Bκ \Bσt

Z sin(κbk ·Ex − κs) − sin(t−β bk ·Ex −t−β s) = Σ l (k) dΩ (A.8) bEv j b Ek bk ·Ex − s 2 Z ≤ |Σbv (bk)|dΩ , (A.9) ρ s Ej Ek where Σ l (k) := |Ek|2 Σ l (Ek). bEv j b Ev j Infraparticle States in QED I 61

For Ex in the set 3 {Ex ∈ R : |Ex| > (1 − ρ)s, 0 < ρ < 1}, we integrate by parts with respect to cosθEk, and observe that the two functions

l d(ΣbEv (bk)) l (k) and j (A.10) ΣbEv j b d cosθEk 1 2 belong to L (S ; dΩEk). This yields Z l (Ek)cos(Ek · x − |Ek|s)d3k (A.11) ΣEv j E Bκ \Bσt Z κ Z sin(|Ek||Ex| + |Ek|s) = − Σ l (k)| d|Ek|dφ (A.12) bEv j b θEk=π Ek σt |Ek||Ex| Z κ Z sin(|Ek||Ex| − |Ek|s) − Σ l (k)| d|Ek|dφ (A.13) bEv j b θEk=0 Ek σt |Ek||Ex| Z d(Σ l (k)) 3 bEv j b sin(Ek ·Ex − |Ek|s) d k − . (A.14) 2 Bκ \Bσt d cosθEk |Ek||Ex| |Ek|

|lnσt | The absolute values of (??), (??), and (??), are all bounded above by O s−ρs , as one easily veriﬁes. This establishes (??), uniformly in Ex ∈ R3. To prove (??), we consider Ex in a set of the form 3 0 0 {Ex ∈ R : (1 − ρ )s > |Ex| > (1 − ρ)s, 0 < ρ < ρ < 1}. (A.15) S We apply integration by parts with respect to |Ek| in (??), (??), and (??) in the case σt . As an example, we get for (??), Z l cos(κ (|Ex| + s)) (??) = Σb (bk)|θ =π dφ (A.16) Ev j Ek κ (|Ex| + s)|Ex| Ek Z −θ l cos(t (|Ex| + s)) − Σb (bk)|θ =π dφ (A.17) Ev j Ek t−θ (|Ex| + s)|Ex| Ek Z κ Z l cos(|Ek||Ex| + |Ek|s) + Σb (bk)|θ =π dφ d|k|. (A.18) S Ev j Ek 2 Ek σt |Ek| |Ex|(|Ex| + s) Since Ex is assumed to be an element of (??), it follows that the bound (??) holds for (??). In the same manner, one obtains a similar bound for (??) and (??). ut Theorem A3 For θ < 1 sufﬁciently close to 1, and s ≥ t, the propagation estimate

σ σ Ex iγ (Ev ,E∇E t ,s) −iE t s (t) χ e σt j PE e PE ψ (A.19) h s j,σt σ σ σ iγ (Ev ,E∇E t ,s) −iE t s (t) − χ (E∇E t )e σt j PE e PE ψ h PE j,σt 1 1 ≤ c | ( )| ν 3ε ln σt (A.20) s t 2 holds, where ν > 0 is independent of ε. Proof Since the detailed proof of a closely related result is given in Theorem A2 of ?, we only sketch the argument. Expressing χh (which we assume to be real) in terms of its Fourier transform χbh, start from the bound 62 T. Chen, J. Frohlich,¨ A. Pizzo

Z σt x σt σt 3 −iEq·E∇E −iq· E iγσ (Ev j,E∇E ,s) −iE s (t) k d q (q)(e PE − e E s )e t PE e PE k (A.21) χbh E ψ j,σt

σt σt Z σt i(E −E )s σt 3 −iEq·E∇E PE E Eq iγσ (Ev j,E∇E ,s) (t) ≤ k d q (q)(e PE − e P+ s )e t PE k (A.22) χbh E ψ j,σt

σt σt Z i(E −E )s x σt 3 PE E Eq −iq· E iγσ (Ev j,E∇E ,s) (t) +k d q (q)e P+ s (e E s − 1)e t PE k. (A.23) χbh E ψ j,σt We split the integration domains of (??) and (??) into the two regions

1−θ 1−θ I+ := {Eq : |Eq| > s } and I− := {Eq : |Eq| ≤ s }. (A.24)

In both (??) and (??), the contribution to the integral from I+ is controlled by the decay properties of χb(Eq), and one easily derives the bound in (??). For the contributions to (??) from the integral over I−, the existence of the gradient of the energy, the Holder¨ property in PE of the gradient, and the decay properties of χb(Eq) are enough to produce the bound in (??). To control (??), we note that the two vectors

σ σ Ex σ Ψ t and Ψ t := e−iEq· s Ψ t (A.25) E Eq bE Eq PE P− s P− s

σt σt belong to the same ﬁber space q , and that, as vectors in Fock space,Ψ andΨ coincide, HPE− E bE Eq PE s P− s i.e.,

Ex −iEq· s σt σt IE Eq (e Ψ ) ≡ IPE(Ψ ). (A.26) P− s PE PE

We split and estimate (??)|I− , i.e., (??) where the integration domain is restricted to I−, by σt σt Z Z i(E q −E )s E σt PE− E PE iγσt (Ev j,∇E ,s) σt 3 3 (??)|I = χh(Eq) e s h e PE Ψb d Pd q (A.27) − b (t) PE PE− Eq I− Γj s σt σt Z Z i(E −E )s σt PE E Eq iγσ (Ev j,E∇E ,s) σ 3 3 − χ (q) e P+ s h e t PE Ψ t d Pd q (A.28) bh E (t) PE PE I− Γj σt σt Z Z i(E q −E )s E σt PE− E PE iγσt (Ev j,∇E ,s) σt 3 3 ≤ χh(Eq) e s h e PE Ψb d Pd q (A.29) b (t) PE PE− Eq I− Γj s σt σt Z Z i(E q −E )s E σt PE− E PE iγσt (Ev j,∇E ,s) σt 3 3 − χh(Eq) e s h e PE Ψ d Pd q b (t) PE PE− Eq I− Γj s σt σt Z Z i(E q −E )s E σt PE− E PE iγσt (Ev j,∇E ,s) σt 3 3 + χh(Eq) e s h e PE Ψ d Pd q (A.30) b (t) PE PE− Eq I− Γj s σt σt E σt Z Z i(E q −E )s iγσt (Ev j,∇E q ,s) PE− E PE PE− E σt 3 3 − (q) e s h q e s d Pd q χh E (t) E E Ψ Eq b P− s PE− I− Γj s σt σt E σt Z Z i(E q −E )s iγσt (Ev j,∇E q ,s) PE− E PE PE− E σt 3 3 + (q) e s h q e s d Pd q χh E (t) E E Ψ Eq b P− s PE− I− Γj s σt σt Z Z i(E −E )s σt PE E Eq iγσ (Ev j,E∇E ,s) σ 3 3 − χ (q) e P+ s h e t PE Ψ t d Pd q . (A.31) bh E (t) PE PE I− Γj The terms (??), (??), and (??) can be bounded by

1 Z "Z # 2 2 σt σt 2 3 3 ( ) ≤ | (q)| |h | kI ( ) − I q ( )k d P d q, ?? χh E (t) PE PE Ψ E E Ψ Eq F b PE P− s PE− I− Γj s (A.32) Infraparticle States in QED I 63

1 " # 2 Z Z E σt iγσt (Ev j,∇E ,s) 2 σt 2 3 3 ( ) ≤ | (q)| | q (h e PE )| kI q ( )k d P d q, ?? χh E (t) ∆ E PE E E Ψ Eq F b s P− s PE− I− Γj s (A.33) and

1   2 Z Z (??) ≤ |χ (q)| |h |2 kI (Ψ σt )k2 d3P d3q, (A.34) bh E  j PE PE E F  I O P − Eq s where

σt σ σ iγ (Ev ,E∇E ,s) iγ (Ev ,E∇E t ,s) iγ (Ev ,E∇E t ,s) σt j Eq σt j PE σt j PE PE− s ∆ Eq (hPEe ) := hPEe − hE Eq e , (A.35) s P− s

Eq Eq Eq j (t) (t), s (t) (t), s (t), s Eq (t) and O Eq := (Γj ∪Γj )\(Γj ∩Γj ), where Γj is the translate by s of the cell Γj . s 1 Using (??), the C −regularity of hPE, and the deﬁnition of I−, one readily shows that the terms (??), (??) satisfy the bound (??), as desired. To estimate (??), we use the inequality

σt σt kI Ψ − I q Ψ k (A.36) PE PE PE− E E Eq F s P− s σt σt σt σt ≤ kI W (E∇E )Ψ − I q W (E∇E )Ψ k (A.37) PE σt PE PE PE− E σt E Eq E Eq F s P− s P− s ∗ σt ∗ σt σt σt +kI q (W (E∇E ) −W (E∇E ))W (E∇E )Ψ k , (A.38) PE− E σt PE σt E Eq σt E Eq E Eq F s P− s P− s P− s where it is clear that

W (E∇Eσt )Ψ σt = Φσt . (A.39) σt PE PE PE

Moreover, we use properties (I 2), (I 5) in Theorem ??, where we recall that

(I 2) Holder¨ regularity in PE ∈ S uniformly in σ ≥ 0 holds in the sense of

σ σ 1 −δ 0 kΦ − Φ k ≤ C 0 |∆PE| 4 (A.40) PE PE+∆PE F δ

and

σ σ 1 −δ 00 |E∇E −E∇E | ≤ C 00 |∆PE| 4 , (A.41) PE PE+∆PE δ

00 0 1 E E E for any 0 < δ < δ < 4 , with P, P + ∆P ∈ S , and where Cδ 0 and Cδ 00 are ﬁnite constants depending on δ 0 and δ 00, respectively. We can bound (??) by use of (??). 64 T. Chen, J. Frohlich,¨ A. Pizzo

In order to bound (??), we recall the deﬁnition of the Weyl operator   Z E∇Eσ 1 3 E ∗ W (E∇Eσ ) := exp α 2 d k P · (Eε b − h.c.) , (A.42) σ PE  ∑ 3 Ek,λ Ek,λ  B \B E 2 λ Λ σ |k| δPE,σ (bk) and we note that  1  Z ! 2 E σt E σt 3 σ 2 (??) ≤ c ∇E − ∇E Rt Rt + d kkb Φ q k PE PE− Eq  ∑ Ek,λ PE− E F  s λ BΛ \Bσt s (A.43) from a simple application of the Schwarz inequality, where

1 3 ! 2 Z d k 1 Rt := = O(|lnσt | 2 ). (A.44) 3 BΛ \Bσt |Ek| Moreover, we have Z 3 σ 2 d kkb Φ k ≤ c|lnσt |, (A.45) ∑ Ek,λ PE− Eq F λ BΛ \Bσt s which is derived similarly as (??). From Holder¨ continuity of E∇Eσ in PE,(??), we obtain a contribution to the upper bound on PE (??) which exhibits a power law decay in s. We conclude that (??) is bounded by (??), as claimed. ut

Remark By a similar procedure, one ﬁnds that for t2 ≥ s ≥ t1, σ σ t2 t2 Ex iγσ (Ev j,E∇E ,s) −iE s (t ) t2 PE PE 1 χh( )e e ψ j,σ s t2 σ σ t2 t2 σt iγσ (Ev j,E∇E ,s) −iE s (t ) 1 |ln(σt )| E 2 t2 PE PE 1 2 − χh(∇E )e e ψ j,σ ≤ c . (A.46) PE t2 sν 3/2 t1 Analogous extensions hold for the estimates in the next theorem. Theorem A4 Both

+∞ " Ex # Z σt S Ex dγσ (Ev j, ,s) eiH s (v ,s) |σs (s) ( ) − b t s Wκ,σt Ej J σt χh t s ds

σ σ iγ (Ev ,E∇E t ,s) −iE t s σ (t) ×e σt j PE e PE (E t + i)ψ ds (A.47) PE j,σt and Ex Z +∞ σ d (v , ,s) σt σt iH t s n γbσt Ej s iγσ (Ev j,E∇E ,s) −iE s σt (t) e (v ,s) e t PE e PE (E + i)ψ Wκ,σt Ej PE j,σt t ds

σt dγσ (Ev j,E∇E ,s) σt σt t PE iγσ (Ev j,E∇E ,s) −iE s σ (t) o − e t PE e PE (E t + i)ψ ds (A.48) PE j,σt ds are bounded by

1 2 − 3ε c |ln(σ )| t 2 , (A.49) tη t σ S Ex dγ (Ev ,E∇E t ,s) σs dγbσt (Ev j, s ,s) σt j PE where η > 0 is ε-independent. J |σt (s), ds , and ds are deﬁned in (??), (??), and (??)–(??), respectively. Infraparticle States in QED I 65

S Proof We recall from (??) that for σs ≥ σt ,

S Z σs σt 1 3 J | (s) = α i[H ,Ex] · ΣEv (Ek) cos(Ek ·Ex − |Ek|s)d k, σt Ej σt B S \Bσ H + i σs t

S = 1 ?? where σs : sθ is the slow cut-off, and from ( ),

Ex iHσt s −iHσt s dγˆσ (Ev j, ,s) σt 1 d{e Ex (s)e } σt t s := α e−iH s h eiH s · ds Hσt + i ds Z · ΣE (Ek) cos(Ek ·E∇Eσt s − |Ek|s)d3k. (A.50) Ev j PE B S \Bσ σs t

S iHσt s For s such that σs ≤ σt the expressions (??) and (??) are identically zero. By unitarity of e S σs and Wσt (Ev j,s), we can replace the part in the integrand of (??) proportional to J |σt (s) by

σ Z iH t s σt 1 Ex 3 σt e Wκ,σ (Ev j,s)αi[H ,Ex] χh( ) · d kΣEv (Ek)cos(Ek ·E∇E s − |Ek|s) t σt Ej PE H + i s B S \Bσ σs t σ σ iγ (Ev ,E∇E t ,s) −iE t s σ (t) ×e σt j PE e PE (E t + i)ψ , (A.51) PE j,σt up to a term which yields an integral bounded in norm by

1 2 − 3ε |ln(σ )| t 2 , (A.52) tη t for some η > 0 and independent of ε. To justify this step, we exploit the fact that the operator 1 i[Hσt , Ex] (A.53) Hσt + i is bounded. Moreover, we are applying the propagation estimate nZ Z o ΣE (Ek)cos(Ek · x − |Ek|s)d3k − ΣE (Ek)cos(Ek ·E∇Eσt s − |Ek|s)d3k Ev j E Ev j PE B S \Bσ B S \Bσ σs t σs t σ σ iγ (Ev ,E∇E t ,s) −iE t s σ (t) 1 1 ·e σt j PE e PE (E t + i)ψ ≤ c |ln(σ )|, (A.54) PE j,σt 1+ν 3ε t s t 2 for some ν > 0, which is similar to (??). To obtain the upper bound, we exploit the fact that due S S −θ σs to the slow cut-off σs = s , θ > 0, in J |σt (s), the upper integration bound in the radial part of the momentum variables vanishes in the limit s → ∞. We note that we have to assume θ < 1 as required in (??), in order to use the result in Lemma ??. Next, we approximate (??) by

σ σ Z iH t s −iH t s 1 dEx(s) Ex(s) 3 σt e Wκ,σ (Ev j,s)α e χh( ) · d kΣEv (Ek)cos(Ek ·E∇E s − |Ek|s) t σt Ej PE H + i ds s B S \Bσ σs t σ σ iγ (Ev ,E∇E t ,s) (t) ·(E t + i)e σt j PE ψ , (A.55) PE j,σt

σ σ where Ex(s) := eiH t sExe−iH t s. To pass from (??) to (??), we have used

dEx(s) 1 1 dEx(s) 1 d[Ex(s), Hσt ] 1 = − , (A.56) ds Hσt + i Hσt + i ds Hσt + i ds Hσt + i 66 T. Chen, J. Frohlich,¨ A. Pizzo and we have noticed that the term containing

1 d[Ex(s), Hσt ] 1 (A.57) Hσt + i ds Hσt + i can be neglected because an integration by parts shows that the corresponding integral is bounded 3ε 1 2 − 2 in norm by tν |ln(σt )| t for some ν > 0 and ε-independent. This uses

Z |ln(σ )| sup d3kΣE (Ek) cos(Ek ·E∇Eσt s − |Ek|s) ≤ t (A.58) Ev j PE O E B S \Bσ s P∈S σs t and

d Z 1 sup d3kΣE (Ek)cos(Ek ·E∇Eσt s − |Ek|s) ≤ , (A.59) Ev j PE O 1+θ E ds B S \Bσ s P∈S σs t which can be derived as in Lemma ??. To bound the integral corresponding to (??), we note that up to a term whose integral is dEx(s) Ex(s) bounded in norm by (??), one can replace ds χh( s ) by

d σt σt eiH sEx (s)e−iH s , (A.60) ds h

Ex where Exh(s) := Exχh( s ), with χh(Ey) deﬁned as in Sect. ??. This is possible because

d σt σt eiH sEx (s)e−iH s (A.61) ds h σt Ex Ex σt = −eiH sEx[ ·E∇χ ( )]e−iH s (A.62) s2 h s σt Ex σt +eiH si[Hσt , Ex]χ ( )e−iH s (A.63) h s σt σt Ex Ex i[H , Ex] σt +eiH s E∇χ ( ) · e−iH s (A.64) s h s 2 σt σt Ex i[H , Ex] Ex σt +eiH s ·E∇χ ( ) e−iH s, (A.65) s 2 h s

dEx(s) Ex(s) where (??) corresponds to ds χh( s ). Moreover, we use the fact that the vector operator i j 1 i[Hσt , x] x x j (Ex ) Hσt +i E is bounded, and apply the propagation estimate (??) to s2 ∇ χh s and to i x ∇ j χ (Ex ) with appropriate modiﬁcations (see (??) and recall that E∇χ (E∇Eσt ) = 0 for PE ∈ s h s h PE supph). We observe that iHσt s −iHσt s Z iHσt s −iHσt s 1 d(e Exh(s)e ) 3 e W , (Ev j,s)α e · d kΣEv (Ek) κ σt σt Ej H + i ds B S \Bσ σs t σ σ σ iγ (Ev ,E∇E t ,s) (t) · cos(Ek ·E∇E t s − |Ek|s)(E t + i)e σt j PE ψ (A.66) PE PE j,σt corresponds to

" Ex # σ dγˆ (v , ,s) E σt iH t s σt Ej s iγσt (Ev j,∇E ,s) (t) e W , (Ev j,s) e PE ψ . (A.67) κ σt ds j,σt

This immediately implies (??). Infraparticle States in QED I 67

To prove (??), we need to control the integral

Z s¯ iHσt s −iHσt s Z iHσt s −iHσt s α d(e Exh(s)e ) 3 e W , (Ev j,s)e · d kΣEv (Ek) κ σt σt Ej t H + i ds B S \Bσ σs t σ σ iγ (Ev ,E∇E t ,s) σ (t) ·cos(Ek ·E∇E t s − |Ek|s)e σt j PE (E t + i)ψ ds (A.68) PE PE j,σt fors ¯ → +∞. An integration by parts with respect to s yields Z iHσt s α 3 e W , (Ev j,s) Exh(s) · d kΣEv (Ek) κ σt σt Ej H + i B S \Bσ σs t σt σt σ −iE s iγσ (Ev j,E∇E ,s) σ (t) s¯ ×cos(Ek ·E∇E t s − |Ek|s)e PE e t PE (E t + i)ψ (A.69) PE PE j,σt t Z s¯ n d iHσt s −iHσt s o iHσt s α − (e Wκ,σt (Ev j,s)e ) e t ds Hσt + i Z ×x (s) · d3kΣE (Ek)cos(Ek ·E∇Eσt s − |Ek|s) E h Ev j PE B S \Bσ σs t σ σ −iE t s iγ (Ev ,E∇E t ,s) σ (t) ×e PE e σt j PE (E t + i)ψ ds (A.70) PE j,σt Z s¯ iHσt s α − e Wκ,σt (Ev j,s) t Hσt + i ( " #) Z σt d 3 σ iγσ (Ev j,E∇E ,s) ×x (s) · d kΣE (Ek)cos(Ek ·E∇E t s − |Ek|s)e t PE E h Ev j PE ds B S \Bσ σs t σ −iE t s σ (t) ×e PE (E t + i)ψ ds (A.71) PE j,σt Here, we notice that

Ex Z Ex Ex (s) = Exχ ( ) = −is E∇χ (Eq)e−iEq· s d3q. (A.72) h h s ch Furthermore, the operator

Z Ex E −iEq· s 3 − i ∇cχh(Eq)e d q (A.73) tends to

Z σ −iEq·E∇E t E PE 3 − i ∇cχh(Eq)e d q (A.74) for s → ∞, if it is applied to the vectors

σ σ iγ (Ev ,E∇E t ,s) −iE t s (t) e σt j PE e PE , (A.75) ψ j,σt or

Z σt σt 3 σ iγσ (Ev j,E∇E ,s) −iE s (t) d kΣE (Ek)cos(Ek ·E∇E t s − |Ek|s)e t PE e PE ψ , (A.76) Ev j PE j,σt B S \Bσ σs t or ( " #) Z σt σt d 3 σ iγσ (Ev j,E∇E ,s) −iE s (t) d kΣE(Ek,v )cos(Ek ·E∇E t s − |Ek|s)e t PE e PE ψ . Ej PE j,σt ds B S \Bσ σs t (A.77) 68 T. Chen, J. Frohlich,¨ A. Pizzo

The rate of convergence of the corresponding expression in (??) is bounded by (??). Therefore, we can replace expressions (??),(??), and (??) by

σ eiH t s (v ,s)α sE∇Eσt (A.78) Wκ,σt Ej PE Z σt σt s¯ 3 σ iγσ (Ev j,E∇E ,s) −iE s (t) · d kΣE (Ek)cos(Ek ·E∇E t s − |Ek|s)e t PE e PE ψ Ev j PE j,σt B S \Bσ t σs t s Z ¯ d σ σ iH t s −iH t s E σt − ds (e Wκ,σt (Ev j,s)e ) α s∇EE (A.79) t ds P Z σt σ 3 iγσ (Ev j,E∇E ,s) (t) · ΣE (Ek)cos(Ek ·E∇E t s − |Ek|s)d ke t PE ψ Ev j PE j,σt B S \Bσ σs t

Z s¯ σ iH t s E σt − e Wκ,σt (Ev j,s)α s∇EE (A.80) t P ( " #) Z σt d σ 3 iγσ (Ev j,E∇E ,s) · ΣE (Ek)cos(Ek ·E∇E t s − |Ek|s)d ke t PE Ev j PE ds B S \Bσ σs t −iEσt s (t) ×e PE ds. ψ j,σt Recalling the deﬁnition of the phase factor in (??)-(??), the sum of the expressions (??), (??), and (??) can be written compactly as

Z s¯ σt σt σt dγσt (Ev j,∇EE,s) iγ (Ev ,E∇E ,s) −iE s (t) iH s P σt j PE PE ds e Wκ,σt (Ev j,s) e e ψ j,σ , (A.81) t ds t after an integration by parts. This implies the asserted bound for (??). ut

Acknowledgements The authors gratefully acknowledge the support and hospitality of the Er- win Schrodinger¨ Institute (ESI) in Vienna in June 2006, where this collaboration was initiated. T.C. was supported by NSF grants DMS-0524909 and DMS-0704031. A. P. was supported by NSF grant DMS-0905988.

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