Infraparticle Scattering States in Non-Relativistic Qed: I

INFRAPARTICLE SCATTERING STATES IN NON-RELATIVISTIC QED: I. THE BLOCH-NORDSIECK PARADIGM

THOMAS CHEN, JURG¨ FROHLICH,¨ AND ALESSANDRO PIZZO

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 infamous 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 a finite number of photons are ill-defined, because, typically, Feynman amplitudes containing vertex corrections exhibit logarithmic infrared divergences; [14, 20]. A pragmatic approach proposed by Jauch and Rohrlich, [19, 25], and by Yennie, Frautschi, and Suura, [29], 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 > 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 , 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 is -independent: Because the transition probability P for an inclusive process is estimated to be O(const.α), the threshold energy cannot be allowed to approach zero, unless ”Bremsstrahlungs”processes (emission of photons) are properly incorporated 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 1 2 T.CHEN, J.FROHLICH,¨ AND A. PIZZO by order in perturbation theory. The works of Chung [12], Kibble [21], and Faddeev and Kulish [13], 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, [2], and extended by Pauli and Fierz, [14], in the late 1930’s. In a seminal paper [2] 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 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 remove some of the major conceptual problems. New puzzles appeared, some of which are related to the problem that Lorentz boosts cannot be unitarily implemented on charged scattering states; see [18]. 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 [28]. Subsequent developments 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 [26], [4]. ii) Corrections to the asymptotic dynamics, as compared to the one in a theory with a positive mass gap; see [3]. iii) Superselection rules pertaining to the space-like asymptotics of the quantized electromagnetic ﬁeld, and connections to Gauss’ law; see [4].

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 [15], [16], 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 [15, 16] have become standard tools in more recent work on non-relativistic QED, and attempts made in [15, 16] 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 Nelson’s model constitutes an active branch of contemporary mathematical physics. In questions relating to INFRAPARTICLE STATES IN QED I. 3 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 [24] by introducing a suitable scattering scheme. This analysis involves spectral results substantially improving those in [16]. It is based on a new multiscale technique developed in [23]. 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 [23, 24] 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 [23, 24]. In a companion paper, [11], 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 [23]. Bogoliubov transformations, proven in [10] to characterize the soft photon clouds in non-relativistic QED, represent an important element in our construction. The proof in [10] uses the uniform bounds on the renormalized electron mass previously established in [9]. We present a detailed definition of the model of non-relativistic QED in Section II. Aspects of infraparticle scattering theory, developed in this paper, are described in Section III. To understand why background radiation parametrized by the asymptotic velocities of the charged particles is expected to be present in all the scattering states, a very useful point of view has been brought to our attention by Morchio and Strocchi. It relies on a classical argument: Consider a current Jµ describing a point-like classical particle of velocity ~v, and 1 charge −α 2 ,

3/2 1/2 (3) 3/2 1/2 (3) Jµ(t, ~y) := ( −4π α δ (~y − ~vt) , 4π α ~vδ (~y − ~vt)) . (I.1)

The Maxwell equations determining the e.m. ﬁelds generated by this current are given by

2 ~ ~ (∂t − ∇ · ∇)Aµ(t, ~y) = Jµ(t, ~y) (I.2) ν ∂µ∂A(t, ~y) = 0 (∂A ≡ ∂νA ) , (I.3) where Aµ(~y, t) is the e.m. potential in an arbitrary gauge that satisﬁes (I.3). (Here and in the rest of this paper, we are using units where c = 1 for the speed of light.) The back-reaction of the electromagnetic ﬁeld on the charged particle is neglected in our considerations.

From (I.2), (I.3), it follows that the ﬁeld Aµ(~y, t) does not have compact support in position space, because of Gauss’ law. A state of the classical system is parameterized by 4 T.CHEN, J.FROHLICH,¨ AND A. PIZZO the data ˙ (~v, Aµ(0, ~y), Aµ(0, ~y)) , (I.4) with (I.3) assumed to be valid at time t = 0. It is of interest to distinguish states or solutions according to the following criterion: We deﬁne a class, C ~vL.W. , of states of the form (I.4) by ˙ the property that Aµ(0, ~y), Aµ(0, ~y) can be written as

~vL.W. Aµ(0, ~y) = Aµ (0, ~y) + hµ(0, ~y) , (I.5)

˙ ˙~vL.W. ˙ Aµ(0, ~y) = Aµ (0, ~y) + hµ(0, ~y) , (I.6) where:

~vL.W. ˙~vL.W. • Aµ (0, ~y), Aµ (0, ~y) are the Cauchy data for the so-called Li´enard-Wiechert (L.W.)-

or causal solution of the equations (I.2), (I.3), with a constant velocity ~v = ~vL.W.. The corresponding electric and magnetic ﬁelds are given by

~n − ~vL.W. E~ ~vL.W. (t, ~y) = −π1/2α1/2[ ] (I.7) γ2 (1 − ~v · ~n)3R2 ret ~vL.W. L.W. ~ ~vL.W. ~ ~vL.W. B (t, ~y) = [~n × E ]ret , (I.8) where 2 −1/2 γ~vL.W. = (1 − ~vL.W.) , (I.9) ~n is the unit vector pointing from the charge to the observer in the point ~y at time t,

R is the distance between the charge and the observer, and [ ]ret implies evaluation at the retarded time. ˙ • hµ(0, ~y) and hµ(0, ~y) are such that 1 (∂ h − ∂ h )(~y, 0) = o( ) , as |~y| → +∞ . (I.10) µ ν ν µ |~y|2 The spatial asymptotics of the ﬁelds is determined by (I.7), (I.8) and characterizes the class C ~vL.W. . Moreover, one can check that the ﬁelds corresponding to an initial condition in

~vL.W. C , but with ~v 6= ~vL.W. in (I.1), (I.2), can be written as the sum of Liénard-Wiechert ~v fields Fµν corresponding to a velocity ~v (that is, by the fields (I.7) and (I.8), but with ~vL.W. replaced by ~v) and a solution of the homogenous Maxwell equations given by

~v Fµν(t, ~y) = ∂µAν(t, ~y) − ∂νAµ(t, ~y) − Fµν(t, ~y) . (I.11)

Hence radiation described by the ~v- and ~vL.W.-dependent ﬁeld Fµν is emitted in any state

~vL.W. in the class C , except when ~v ≡ ~vL.W.. Notice that, at ﬁxed time t, the ﬁelds Fµν do 1 not have compact support in position space. They decay like |~y|2 . This is because in the decomposition ~v ∂µAν(t, ~y) − ∂νAµ(t, ~y) = Fµν(t, ~y) + Fµν(t, ~y) , (I.12) INFRAPARTICLE STATES IN QED I. 5

~vL.W. Fµν(t, ~y) t=0 restores the Cauchy data appropriate for a state in C . The state considered here corresponds to a velocity ~v of the particle and Cauchy data

~vL.W. Aµ(0, ~y) = Aµ (0, ~y) + hµ(0, ~y) , (I.13)

˙ ˙~vL.W. ˙ Aµ(0, ~y) = Aµ (0, ~y) + hµ(0, ~y) , (I.14)

~v differing from those defining Fµν(0, ~y). In the Coulomb gauge the free field in (I.11) has the form

as 1 F0i(t, ~y) = −∂0A (t, ~y) + o( ) (I.15) i |~y|2

as as 1 Fij(t, ~y) = −∂iA (t, ~y) + ∂jA (t, ~y) + o( ) (I.16) j i |~y|2 where Z 3 ~v · ~ε ∗ as. 1 X d k n ~k,λ −i~k·~y+i|~k|t o A~ (t, ~y) := α 2 ~ε e + c.c. (I.17) q 3 ~k,λ ~ 2 ˆ λ |~k| |k| (1 − ~v · k) ∗ Z 3 ~vL.W. · ~ε 1 X d k n ~k,λ −i~k·~y+i|~k|t o − α 2 ~ε e + c.c. . q 3 ~k,λ ~ 2 ˆ λ |~k| |k| (1 − ~vL.W. · k)

~vL.W. The classes C of states of the classical system (indexed by ~vL.W.) correspond to superselection sectors in 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 ~vL.W. = 0. This implies that, in the Fock representation, an asymptotic background radiation must be expected for all values ~v 6= 0 of the asymptotic velocity of the electron. In particular, after replacing the classical velocity ~v with the spectral values of the quantum operators ~vout/in (where ~vout/in is the asymptotic velocity of an outgoing or incoming asymptotic electron, respectively), the background ﬁeld (I.17), with ~vL.W. = 0, corresponds to the background radiation described by the coherent (non-Fock) representations of the asymptotic photon algebra labeled by ~vout/in. This is explained in detail in Section VI; see also Section III.5. 6 T.CHEN, J.FROHLICH,¨ AND 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 convenience, 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 = C Ω , (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 polar- izations (or helicities) of a photon. In (II.3), SN denotes the orthogonal projection onto NN 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. It is convenient to represent the Hilbert space H as the space of square-integrable wave functions on the electron position space R3 with values in the photon Fock space F, i.e.,

∼ 2 3 H = L (R ; F) . (II.5)

In this paper, we use units such that Planck’s constant ~, 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 − i∇~ + α1/2A~(~x) 2 H := ~x + Hf . (II.6) 2 The multiplication operators ~x ∈ R3 correspond to the position of the electron. The electron ~ ∼ momentum operator is given by ~p = −i∇~x; α = 1/137 is the feinstructure constant (which, in this paper, is treated as a small parameter), A~(~x) denotes the (ultraviolet regularized) INFRAPARTICLE STATES IN QED I. 7 vector potential of the transverse modes of the quantized electromagnetic ﬁeld at the point ~x (the electron position) in the Coulomb gauge, ~ ~ ∇~x · A(~x) = 0 . (II.7)

Hf is the Hamiltonian of the quantized, free electromagnetic ﬁeld, given by X Z Hf := d3k |~k| a∗ a , (II.8) ~k,λ ~k,λ λ=± where a∗ and a are the usual photon creation- and annihilation operators, which satisfy ~k,λ ~k,λ the canonical commutation relations

∗ 0 [a , a ] = δ 0 δ(~k − ~k ) , (II.9) ~k,λ ~k0,λ0 λλ [a# , a# ] = 0 , (II.10) ~k,λ ~k0,λ0 with a# = a or a∗. The vacuum vector Ω obeys the condition

a~k,λ Ω = 0 , (II.11) ~ ~ 0 3 0 for all k, k ∈ R and λ, λ ∈ Z2 ≡ {+, −}. The quantized electromagnetic vector potential is given by Z 3 X d k ~ ~ A~(~y) := ~ε e−ik·~ya∗ + ~ε ∗ eik·~ya , (II.12) q ~k,λ ~k,λ ~k,λ ~k,λ λ=± BΛ |~k|

3 where ~ε~k,−, ~ε~k,+ are photon polarization vectors, i.e., two unit vectors in R ⊗ C satisfying ~ε ∗ · ~ε = δ , ~k · ~ε = 0 , (II.13) ~k,λ ~k,µ λµ ~k,λ ~ for λ, µ = ±. The equation k · ~ε~k,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 (II.12) 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 Hamitonian H is selfadjoint on D(H0), the domain of definition of the operator (−i∇~ )2 H := ~x + Hf . (II.14) 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

P~ := ~p + P~ f , (II.15) 8 T.CHEN, J.FROHLICH,¨ AND A. PIZZO ~ where ~p = −i∇~x is the momentum operator for the electron, and X Z P~ f := d3k ~k a∗ a (II.16) ~k,λ ~k,λ λ=± is the momentum operator for the radiation ﬁeld. ~ The operators H and P are essentially selfadjoint on the domain D(H0), and since the dynamics is invariant under translations, they commute: [H, P~ ] = 0. The Hilbert space H can be decomposed on the joint spectrum, R3, of the component-operators of P~ . Their spectral measure is absolutely continuous with respect to Lebesgue measure, Z ⊕ 3 H := HP~ d P, (II.17) where each ﬁber space HP~ is a copy of Fock space F.

Remark: Throughout this paper, the symbol P~ 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.

To each ﬁber space HP~ there corresponds an isomorphism

b IP~ : HP~ −→ F , (II.18)

b where F is the Fock space corresponding to the annihilation- and creation operators b~k,λ, b∗ , where b is given by ei~k·~xa , and b∗ by e−i~k·~xa∗ , with vacuum Ω = I (eiP~ ·~x), where ~k,λ ~k,λ ~k,λ ~k,λ ~k,λ f P~

~x is the electron position. To deﬁne IP~ more precisely, we consider an (improper) vector ψ ∈ H with a deﬁnite total momentum, which describes an electron and n photons (f1,...,fn;P~ ) P~ ~ ~ in a product state. Its wave function, in the variables (~x; k1,..., kn; λ1, . . . , λn), is given by

~ ~ ~ 1 X ei(P −k1−···−kn)·~x f (~k ; λ ) ··· f (~k ; λ ) , (II.19) n! p(1) 1 p(1) p(n) n p(n) p∈Pn

Pn being the group of permutations of n elements. The isomorphism IP~ acts by way of ~ ~ ~ 1 X I ei(P −k1−···−kn)·~x f (~k ; λ ) ··· f (~k ; λ ) = (II.20) P~ n! p(1) 1 p(1) p(n) n p(n) Pn 1 Z = √ d3k0 . . . d3k0 f (~k0 ; λ ) ··· f (~k0 ; λ ) b∗ ··· b∗ Ω . (II.21) 1 n 1 1 1 n n n ~k0 ,λ ~k0 ,λ f n! 1 1 n n In the sequel, we will also use the notation f λ(~k) for f(~k, λ).

The Hamiltonian H maps each ﬁber space HP~ into itself, i.e., it can be written as Z 3 H = HP~ d P, (II.22) INFRAPARTICLE STATES IN QED I. 9 where

HP~ : HP~ −→ HP~ . (II.23) Written in terms of the operators b , b∗ , and of the variable P~ , the fiber Hamiltonian H ~k,λ ~k,λ P~ has the form P~ − P~ f + α1/2A~2 H := + Hf , (II.24) P~ 2 where X Z P~ f := d3k ~k b∗ b , (II.25) ~k,λ ~k,λ λ X Z Hf := d3k |~k| b∗ b , (II.26) ~k,λ ~k,λ λ and X Z d3k A~ := b∗ ~ε + ~ε ∗ b . (II.27) q ~k,λ ~k,λ ~k,λ ~k,λ λ BΛ |~k| Let 1 S := { P~ ∈ 3 : |P~ | < } . (II.28) R 3 In order to give precise meaning to the constructions used in this work, we restrict the total momentum P~ 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 P~ to S guarantees that the propagation speed of a dressed electron is strictly smaller than the speed of light. We start by studying a regularized fiber Hamiltonian given by P~ − P~ f + α1/2A~σ2 Hσ := + Hf (II.29) P~ 2 ~ acting on the fiber space HP~ , for P ∈ S, where X Z d3k A~σ := b∗ ~ε + ~ε ∗ b , (II.30) q ~k,λ ~k,λ ~k,λ ~k,λ λ BΛ\Bσ |~k| and where Bσ is a ball of radius σ.

Remark: In a companion paper [11], we construct dressed one-electron states of fixed momentum given by the ground state vectors Ψσj of the Hamiltonians Hσj , and we compare P~ P~ σ σ ground state vectors Ψσj ,Ψ j0 corresponding to different fiber Hamiltonians Hσj , H j0 with P~ P~ 0 P~ P~ 0 P~ 6= P~ 0. We compare these ground state vectors as vectors in the Fock space F b. In the sequel, we use the expression σ kΨσj − Ψ j0 k (II.31) P~ P~ 0 F 10 T.CHEN, J.FROHLICH,¨ AND A. PIZZO as an abbreviation for σ kI (Ψσj ) − I (Ψ j0 )k ; (II.32) P~ P~ P~ 0 P~ 0 F k · k stands for the Fock norm. Höldercontinuity properties of Ψσ in σ and in P~ are proven F P~ in [11]. These properties play a crucial role in the present paper.

II.1. Summary of contents. In Section III, 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 Sections IV and V, ψh,κ(t) is shown to out/in converge 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 [24]. 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 Section III.1), 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 Section VI, we construct the scattering subspaces Hout/in. Vectors in these subspaces are obtained from certain subspaces, H˚out/in, by applying ”hard” asymptotic photon out/in out/in creation operators. These spaces carry representations of the algebras 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 out/in the infrared properties of the representation of Aph identified by Bloch and Nordsieck in their classic paper, [2]. In a companion paper [11], we establish the main spectral ingredients for the construction and convergence of the vectors {ΨP~ ,σ}, as σ tends to 0. These results are obtained with the help of a new multiscale method introduced in [23], to which we refer the reader for some details of the proofs. In the Appendix, we prove some technical results used in the proofs. INFRAPARTICLE STATES IN QED I. 11

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

Hσ ψσ = Eσ(P~ ) ψσ , (III.1) where Hσ = R ⊕ Hσ d3P , and Eσ(P~ ) is a function of the vector operator P~ ; Eσ(P~ ) is P~ the electron energy function defined more precisely in Section III.1. Since in our model non-relativistic matter is coupled to a relativistic field, the form of Eσ(P~ ) is not fixed by symmetry, except for rotation invariance. Furthermore, the solutions of (III.1) must be restricted to vectors of the form Z σ ~ σ 3 ψ (h) = h(P )ΨP~ d P, (III.2) where the support of h is contained in a ball centered at P~ = 0, chosen such that |∇~ Eσ(P~ )| < 1, as a function of P~ . This requirement imposes the constraint that the maximal 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. The guiding principle motivating our analysis of limiting or improper one-particle states, ψσ, 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 en- countered in the scattering states. The analysis of Bloch and Nordsieck, [2], 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 ψ 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 12 T.CHEN, J.FROHLICH,¨ AND A. PIZZO infinitely many asymptotic free photons, in accordance with the observations sketched in (I.1) – (I.17). Our goal in this paper is to translate this physical picture into rigorous mathematics, following suggestions made in [15] and methods developed in [23, 24, 9, 10].

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 [11], and summarized in Theorem III.1 below; (they are analogous to those used in the analysis of Nelson’s model in [24]). We deﬁne the energy of a dressed one-electron state of momentum P~ by

σ σ σ=0 EP~ = inf specHP~ ,EP~ = inf specHP~ = EP~ . (III.3) We refer to Eσ as the ground state energy of the ﬁber Hamiltonian Hσ . We assume that the P~ P~ ﬁnestructure constant α is so small that

~ σ |∇EP~ | < νmax < 1 (III.4) ~ ~ 3 ~ 1 for all P ∈ S := {P ∈ R : |P | < 3 }, for some constant νmax < 1, uniformly in σ. Corresponding to ∇~ Eσ , we introduce a Weyl operator P~ Z ∇~ Eσ σ 1 X 3 P~ ∗ W (∇~ E ) := exp α 2 d k · (~ε b − h.c.) , (III.5) σ P~ 3 ~k,λ ~k,λ ~ 2 λ BΛ\Bσ |k| δP~ ,σ(bk) where ~ ~ σ k δ ~ (bk) := 1 − ∇E ~ · , (III.6) P ,σ P |~k| acting on HP~ , which is unitary for σ > 0. We consider the transformed ﬁber Hamiltonian σ ~ σ σ ∗ ~ σ KP~ := Wσ(∇EP~ )HP~ Wσ (∇EP~ ) . (III.7) We note that conjugation by W (∇~ Eσ ) acts on the creation- and annihilation operators as σ P~ a linear Bogoliubov transformation (translation) 1 (~k) W (∇~ Eσ ) b# W ∗(∇~ Eσ ) = b# − α1/2 σ,Λ ∇~ Eσ · ~ε # , (III.8) σ P~ ~k,λ σ P~ ~k,λ 3 P~ ~k,λ ~ 2 |k| δP~ ,σ(bk) ~ where 1σ,Λ(k) stands for the characteristic function of the set BΛ \Bσ. Our methods rely on proving regularity properties in σ and P~ of the ground state vector, Φσ , and of the P~ ground state energy, Eσ , of Kσ . These regularity properties are summarized in the following P~ P~ theorem, which is the main result of the companion paper [11].

Theorem III.1. For P~ ∈ S and for α > 0 suﬃciently small, the following statements hold. INFRAPARTICLE STATES IN QED I. 13

( 1) The energy Eσ is a simple eigenvalue of the operator Kσ on F b. Let B := {~k ∈ I P~ P~ σ 3 ~ 2 3 R | |k| ≤ σ}, and let Fσ denote the Fock space over L ((R \Bσ) × Z2). Likewise, σ 2 b σ we define F0 to be the Fock space over L (Bσ × Z2); hence F = Fσ ⊗ F0 . On Fσ, the operator Kσ has a spectral gap of size ρ−σ or larger, separating Eσ from the rest P~ P~ of its spectrum, for some constant ρ− (depending on α), with 0 < ρ− < 1. The contour ρ−σ γ := {z ∈ ||z − Eσ | = } , σ > 0 (III.9) C P~ 2 bounds a disc which intersects the spectrum of Kσ in only one point, {Eσ }. The P~ P~ ground state vectors of the operators Kσ are given by P~ 1 R 1 2π i γ Kσ −z dz Ωf σ P~ ΦP~ := 1 R 1 (III.10) k 2π i γ Kσ −z dz Ωf kF P~ b and converge strongly to a non-zero vector ΦP~ ∈ F , in the limit σ → 0. The rate of 1 (1−δ) convergence is of order σ 2 , for any 0 < δ < 1. The dependence of the ground state energies Eσ of the fiber Hamiltonians Kσ on P~ P~ the infrared cutoff σ is characterized by the following estimates.

σ σ0 | EP~ − EP~ | ≤ O(σ) , (III.11) and

0 1 ~ σ ~ σ 2 (1−δ) | ∇EP~ − ∇EP~ | ≤ O(σ ) , (III.12) for any 0 < δ < 1, with σ > σ0 > 0. (I 2) The following H¨olderregularity properties in P~ ∈ S hold uniformly in σ ≥ 0:

σ σ 1 −δ0 0 ~ 4 kΦP~ − ΦP~ +∆P~ kF ≤ Cδ |∆P | (III.13) and

σ σ 1 −δ00 ~ ~ 00 ~ 4 |∇EP~ − ∇EP~ +∆P~ | ≤ Cδ |∆P | , (III.14)

00 0 1 ~ ~ ~ 0 00 for 0 < δ < δ < 4 , with P, P + ∆P ∈ S, where Cδ and Cδ are ﬁnite constants depending on δ0 and δ00, respectively.

(I 3) Given a positive number νmin, there are numbers rα = νmin + O(α) > 0 and νmax < 1 ~ such that, for P ∈ S \ Brα and for α suﬃciently small,

~ σ 1 > νmax ≥ |∇EP~ | ≥ νmin > 0 , (III.15) uniformly in σ. 14 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

(I 4) For P~ ∈ S and for any ~k 6= 0, the following inequality holds uniformly in σ, for α small enough: Eσ > Eσ − C |~k| , (III.16) P~ −~k P~ α where Eσ := inf spec H and 1 < C < 1, with C → 1 as α → 0. P~ −~k P~ −~k 3 α α 3 (I 5) For P~ ∈ S, one has that ~ σ 1/2 1σ,Λ(k) k b~ Ψ ~ k ≤ C α , (III.17) k,λ P |~k|3/2 where Ψσ is the ground state of Hσ ; see Lemma 6.1 of [10] which can be extended to P~ P~ ~k ∈ R3 using (I 4).

Detailed proofs of Theorem III.1 based on results in [23, 10] are given in [11].

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 [24], (see also [15]). In the context of the present paper, our task is to give a mathematically rigorous meaning to the formal expression

out iHt −iHσt σ Φκ (h) := lim lim e Wσ(~v(t), t) e ψ (h) , (III.18) t→∞ σ→0 where ~ ~ Z 3 ~v(t) ·{~ε a∗ e−i|k|t − ~ε ∗ a ei|k|t} 1 d k ~k,λ ~k,λ ~k,λ ~k,λ W (~v(t), t) := exp α 2 . σ q ~ Bκ\Bσ |~k| |k|(1 − bk · ~v(t)) The operator ~v(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 the operator ~ ~ ∇E(P ), applied to the (non-Fock) vectors ΨP~ . This can be seen by ﬁrst 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 −iHσt e Wσ(~v(t), t) e (III.19) is an interpolating operator used in the L.S.Z. (Lehmann-Symanzik-Zimmermann) approach to scattering theory for the electromagnetic field. In the definition of this operator the photon wave functions are evolved backwards in time with the free evolution, and the photon INFRAPARTICLE STATES IN QED I. 15 creation- and annihilation operators are evolved forward in time with the interacting time evolution. Moreover, the photon wave functions in (III.19) coincide with the wave functions in the Weyl operator W (∇~ Eσ ) defined in (III.5), after replacing the operator ∇~ E(P~ ) by the σ P~ operator ~v(t). We stress that, while the Weyl operator W (∇~ Eσ ) leaves the fiber spaces H σ P~ P~ ∗ invariant, the Weyl operator Wσ(~v(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 ∇~ Eσ(P~ ), as t → ∞ and σ → 0, two key ideas used to make (III.18) precise are to render the infrared cut-off time-dependent, ~ 3 ~ 1 with σt → 0, as t → ∞, and to discretize the ball S = {P ∈ R | |P | < 3 }, with a grid size decreasing in time t. This discretization also applies to the velocity operator ~v(t) in expression (III.18). 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 tuning the convergence rates of σt and of the discretization of S. Our sequence of approximate infraparticle scattering states is defined as follows:

i) We consider a wave function h with support in a region R contained in S\Brα ; (see condition (I 3) in Theorem III.1). We introduce a time-dependent cell partition G (t) of R. This partition is constructed as follows: L 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 ) ≤ t < (2 ) , (III.20)

for some > 0 to be ﬁxed later. Thus, the total number of cells in G (t) is N(t) = 23n, (t) th where n = blog2 t c;(bxc extracts the integer part of x). By Gj , we denote the j cell of the partition G (t).

ii) For each cell, we consider a one-particle state of the Hamiltonian Hσt Z (t) ~ σt 3 ψj,σ := h(P )Ψ d P (III.21) t (t) P~ Gj where ~ 1 • h(P ) ∈ C0 (S\Brα ), with supp h ⊆ R; 16 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

−β • σt := t , for some exponent β (> 1) to be ﬁxed later; • In (III.21), the ground state vector, Ψσt , of Hσt is deﬁned by P~ P~ Ψσt := W ∗ (∇Eσt )Φσt , (III.22) P~ σt P~ P~ where Φσt is the ground state of Kσt ; (see Theorem III.1). P~ P~ (t) iii) With each cell Gj we associate a soft-photon cloud described by the following ”LSZ (Lehmann-Symanzik-Zimmermann) Weyl operator”

iHt −iHσt t e Wσt (~vj, t) e , (III.23) where ~ ~ Z 3 ~v ·{~ε a∗ e−i|k|t − ~ε ∗ a ei|k|t} 1 d k j ~k,λ ~k,λ ~k,λ ~k,λ W (~v , t) := exp α 2 . (III.24) σt j q B \B |~k|(1 − k · ~v ) κ σt |~k| b j • Here κ, with 0 < κ ≤ Λ, is an arbitrary (but ﬁxed) soft-photon energy threshold.

~ σt ~ ∗ • ~vj ≡ ∇E (Pj ) is the c-number vector corresponding to the value of the ”veloc- (t) ~ σt ~ ~ ∗ ity” ∇E (P ) in the center, Pj , of the cell Gj .

iv) For each cell, we consider a time-dependent phase factor

σt iγσ (~vj ,∇~ E ,t) e t P~ , (III.25)

with Z t Z γ (~v , ∇~ Eσt , t) := − α ∇~ Eσt · Σ~ (~k) cos(~k · ∇~ Eσt τ − |~k|τ) d3k dτ , (III.26) σt j P~ P~ ~vj P~ 1 B S \Bσ στ t and l l0 l ~ X k k l0 1 Σ~v (k) := 2 (δl,l0 − ) vj . (III.27) j ~ 2 ~ 2 l0 |k| |k| (1 − bk · ~vj) S −θ Here, στ := τ , and the exponent 0 < θ < 1 will be chosen later. Note that, in (III.25), (III.26), ∇~ Eσt is interpreted as an operator. P~

v) The approximate scattering state at time t is given by the expression N(t) σt σt iHt X iγσ (~vj ,∇~ E ,t) −iE t (t) t P~ P~ ψh,κ(t) := e Wσt (~vj, t) e e ψj,σt , (III.28) j=1 where N(t) is the number of cells in G (t). σt iγσ (~vj ,∇~ E ,t) The role played by the phase factor e t P~ 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 INFRAPARTICLE STATES IN QED I. 17 convergent; (see Section III.3).

III.3. Statement of the main result. The main result of this paper is Theorem III.2, below, from which the asymptotic picture described in Section III.5, below, emerges. It relies on the assumptions summarized in the following hypothesis.

Main Assumption III.1. The following assumptions hold throughout this paper:

(1) The conserved momentum P~ takes values in S; see (II.28).

(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. 1, and rα is introduced in (III.15).

L Figure 1. R can be described as a union of cubes with sides of length 2n0 , for some n0 < ∞. 18 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

Theorem III.2. Given the Main Assumption III.1, the following holds: There exist positive real numbers β > 1, θ < 1 and > 0 such that the limit

(out) s − lim ψh,κ(t) =: ψ (III.29) t→+∞ h,κ (out) 2 R ~ 2 3 exists as a vector in H, and kψh,κ k = |h(P )| d P . Furthermore, the rate of convergence is at least of order t−ρ0 , for some ρ0 > 0.

We note that this result corresponds to Theorem 3.1 of [24] 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 Section III.4, we outline the key mechanisms responsible for the convergence of the approximating vectors ψh,κ(t), as t → ∞. We note that, in (III.29), 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 (III.26)); • the rate t− 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 Section IV. • The exponent , 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. INFRAPARTICLE STATES IN QED I. 19

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 → ±∞. Among other things, we will prove that Z ~ 2 3 1 lim kψh,κ(t)k = khkL2 := ( |h(P )| d P ) 2 . (III.30) t→∞

From its deﬁnition, see (III.28), one sees that the square of the norm of the vector ψh,κ(t) involves a double sum over cells of the partitions G (t), i.e., N(t) σt σt σt σt 2 X D iγσ (~vl,∇E ,t) −iE t (t) ∗ iγσ (~vj ,∇E ,t) −iE t (t) E kψ (t)k = e t P~ e P~ ψ , W (~v , t) W (~v , t) e t P~ e P~ ψ , h,κ l,σt σt l σt j j,σt l,j=1 (III.31) where the individual terms, labeled by (l, j), are inner products between vectors labeled by (t) (t) (t) cells Gl and Gj of G . A heuristic argument to see where (III.30) comes from is as follows. Assuming that

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

iHt −iHσt σ out/in lim lim e Wσ(~v(t), t) e ψ (h) = W (~v(±∞)) ψ(h) , (III.32) t→±∞ σ→0 where out/in ∗ ∗ out/in Z 3 ~v(±∞) ·{~ε~ a − ~ε a } out/in 1 d k k,λ ~k,λ ~k,λ ~k,λ W (~v(±∞)) := exp α 2 , q ~ Bκ |~k| |k|(1 − bk · ~v(±∞))

and aout/in ∗ , aout/in are the creation- and annihilation operators of the asymptotic ~k,λ ~k,λ photons; • the operators ~v(±∞) commute with the algebra of asymptotic creation- and annihilation operators {aout/in ∗ , aout/in}; (this can be expected to be a consequence of ~k,λ ~k,λ asymptotic decoupling of the photon dynamics from the dynamics of the electron); • the restriction of the asymptotic velocity operators, ~v(±∞), to the improper dressed one-electron state is given by the operator ∇~ E(P~ ), i.e., ~ ~ ~v(±∞)ΨP~ ≡ ∇E(P )ΨP~ ; (III.33) then, the two vectors

σt σt σt σt iγσ (~vj ,∇E ,t) −iE t (t) iγσ (~vl,∇E ,t) −iE t (t) W (~v , t) e t P~ e P~ ψ and W (~v , t) e t P~ e P~ ψ (III.34) σt j j,σt σt l l,σt 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 (III.31) are the only ones 2 that survive in the limit t → ∞. The fact that their sum converges to khkL2 is comparatively easy to prove. 20 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

A mathematically precise formulation of this mechanism is presented in Section IV. In Section IV.1, part A., the analysis of the scalar products between the cell vectors in (III.34) is reduced to the study of an ODE. To prove (III.30), we invoke the following properties of the one-particle states ψ(t) and ψ(t) located in the j-th and l-th cell: j,σt l,σt • Their spectral supports with respect to the momentum operator P~ are disjoint up to sets of measure zero. • They are vacua for asymptotic annihilation operators, as long as an infrared cut-off λ σt for a fixed time t is imposed: For Schwartz test functions g , we define Z out/in iHσt s X λ ~ i|~k|s −iHσt s 3 aσ (g) := lim e a~ g (k) e e d k , (III.35) t s→±∞ k,λ λ on the domain of Hσt . An important step in the proof of (III.30) is to control the decay in time of the off- diagonal terms. After completion of this step, one can choose the rate, t−, 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 Section IV.1, part B., where we invoke Cook’s argument and analyze the decay in time s of

σ σt σt d iH t s iγσ (~vj ,∇~ E ,s) −iE s (t) e W (~v , s) e t P~ e P~ ψ (III.36) ds σt j j,σt σ σt σt iH t s σ iγσ (~vj ,∇~ E ,s) −iE s (t) t t P~ P~ = i e [HI , Wσt (~vj, s)] e e ψj,σt (III.37)

σ ~ σt σt iH t s dγσt (~vj, ∇EP~ , s) iγσ (~vj ,∇E ,s) −iE s (t) + i e W (~v , s) e t P~ e P~ ψ , (III.38) σt j ds j,σt for a fixed infrared cut-off σt, and a fixed partition. As we will show, the term in (III.37) can be written (up to a unitary operator) as Z Z 1 n o (t) 2 3 ~ ~ 3 α d y Jσt (t, ~y) Σ~vj (~q) cos(~q · ~y − |~q|s) d q ψj,σt (III.39) Bκ\Bσt ~ plus subleading terms, where Jσt (t, ~y) is essentially the electron current, which is proportional to the velocity operator

1 σt 2 ~ i [H , ~x] = ~p + α Aσt (~x) . (III.40) In (III.39), the electron current is smeared out with the vector function Z ~ 3 ~gt(s, ~y) := Σ~vj (~q) cos(~q · ~y − |~q|s) d q , (III.41) Bκ\Bσt which solves the wave equation

s,~y ~gt(s, ~y) = 0 , (III.42) INFRAPARTICLE STATES IN QED I. 21

(t) and is then applied to the one-particle state ψj,σt . Because of the dispersive 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 ~gt(s, ~y) that solves (III.42) propagates along the light cone, and −1 sup 3 |~g (s, ~y)| decays in time like s , while a much faster decay is observed when ~y∈R t ~y is restricted to the interior of the light cone. ~ (t) ii) Because of the support in P of the vector ψj,σt , the propagation of the electron current in (III.37) 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 (III.37) and to complete our argument. An important reﬁnement of this reasoning process, involving the term (III.38), 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

σt σt ~x iγσ (~vj ,∇~ E ,s) −iE s (t) χ ( ) e t P~ e P~ ψ h s j,σt σt σt σ iγσ (~vj ,∇~ E ,s) −iE s (t) − χ (∇~ E t ) e t P~ e P~ ψ h P~ j,σt 1 1 ≤ ν 3 | ln(σt)| , (III.43) s t 2 as s → ∞, where ν > 0 is independent of . Using result (I 3) of Theorem III.1, and our assumption on the support of h formulated in point ii) of Section III.2, 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 ~ 3 ~gt(s, ~y) := Σ~vj (~q) cos(~q · ~y − |~q|s) d q , (III.44) Bκ\Bσt on t, which cannot be neglected even if ~y is in the interior of the light cone. In order to exhibit the desired decay, it is necessary to split ~gt(s, ~y) into two pieces, Z ~ 3 Σ~vj (~q) cos(~q · ~y − |~q|s)d q (III.45) Bκ\B S σs and Z ~ 3 Σ~vj (~q) cos(~q · ~y − |~q|s) d q (III.46) B S \Bσ σs t S S −θ for s such that σs > σt, where σs = s , with 0 < θ < 1. (The same procedure will also be used in (III.49), below.) The function (III.45) has good decay properties inside the light cone. Expression (III.39), with ~gt(s, ~y) replaced by (III.45), can be controlled by standard 22 T.CHEN, J.FROHLICH,¨ AND A. PIZZO dispersive estimates. The other contribution, proportional to (III.46), is in principle singular in the infrared region, but is canceled by (III.38). This can be seen by using a propagation estimate similar to (III.43). This strategy has been designed in [24]. However, because of the vector nature of the interaction in non-relativistic QED, the cancellation in our proof is technically more subtle than the one in [24]. 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.47) where the three terms on the r.h.s. correspond to

(t2) (t1) I) changing the partition G → G in ψh,κ(t2):

(t2) (t1) ∆ψ(t2, σt2 , G → G ) N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t2) −iE t2 (t1) 2 t2 P~ P~ := e Wσt (~vj, t2) e e ψ (III.48) 2 j,σt2 j=1 N(t ) 1 σ σ t2 t2 iHt X X iγσ (~v ,∇~ E ,t2) −iE t2 (t2) 2 t2 l(j) P~ P~ − e Wσt (~vl(j), t2)e e ψ , 2 l(j),σt2 j=1 l(j)

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

σ σ ~ t2 ~ t1 ~vl(j) ≡ ∇E ~ ∗ and ~vj ≡ ∇E ~ ∗ ; Pl(j) Pj

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

infrared cut-oﬀ σt2 :

(t1) ∆ψ(t2 → t1, σt2 , G ) N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t2 P~ P~ := e Wσt (~vj, t1) e e ψ (III.49) 2 j,σt2 j=1 N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t2) −iE t2 (t1) 2 t2 P~ P~ − e Wσt (~vj, t2) e e ψ ; 2 j,σt2 j=1

and, ﬁnally, INFRAPARTICLE STATES IN QED I. 23

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

(t1) ∆ψ(t1, σt2 → σt1 , G ) N(t ) 1 σ σ t1 t1 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t1 P~ P~ := e Wσt (~vj, t1)e e ψ (III.50) 1 j,σt1 j=1 N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t2 P~ P~ − e Wσt (~vj, t1)e e ψ . 2 j,σt2 j=1 It is important to take these three steps in the order indicated above.

(t2) (t1) In step I), the size of k∆ψ(t2, σt2 , G → G )k in (III.48) 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

~vl(j) − ~vj .

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 ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t2 P~ P~ e Wσt (~vj, t1) e e ψ (III.51) 2 j,σt2 j=1 as N(t ) 1 σ σ t2 t2 iHt X ∗ σt σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 ~ 2 t2 P~ P~ e Wσt (~vj, t1) W (∇E ) Wσt (∇E ) e e ψ . (III.52) 2 σt2 P~ 2 P~ j,σt2 j=1

(t1) (t1) The term in (III.52) corresponding to the cell Gj of G can then be obtained by acting with the “dressing operator”

σt σt −iE 2 t iHt1 ∗ ~ 2 ~ 1 e Wσ (~vj, t1) W (∇E ) e P , (III.53) t2 σt2 P~ on the “infrared-regular” vector

Z σ (t1) ~ t2 3 Φj,σt := h(P )Φ ~ d P (III.54) 2 (t1) P Gj corresponding to the vectors Φσ := W ∗(∇~ Eσ )Ψσ deﬁned in (III.22), for all j. The advantage P~ σ P~ P~ of (III.52) over (III.51) is that the vector Φ(t1) inherits the regularity properties of Φσ j,σt2 P~ (t1) described in Theorem III.1. In particular, the vectors Φ converge strongly, as σt → 0, j,σt2 2 and the vector e−i~q·~xΦ(t1) (III.55) j,σt2 24 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

depends on ~q in a strongly Hölder 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 (III.53). Our analysis of the strong convergence of the sequence of approximating vectors cul- minates in the estimate

2 ρ kψh,κ(t2) − ψh,κ(t1)k ≤ O (ln(t2)) /t1 , (III.56) for some ρ > 0. By telescoping, this bound suﬃces to prove Theorem III.2.

(out/in) III.5. A space of scattering states. We use the asymptotic states ψh,κ to construct out/in a subspace, H˚κ , of scattering states invariant under space-time translations, and with a photon energy threshold κ, n o ˚out/in _ out/in ~ 1 3 Hκ := ψh,κ (τ,~a): h(P ) ∈ C0 (S\Brα ) , τ ∈ R, ~a ∈ R , (III.57) where out/in −i~a·P~ −iHτ out ψh,κ (τ,~a) ≡ e e ψh,κ . (III.58) 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 H˚κ by adding “hard” photons, i.e.,

n _ out/in ˆ o Hout/in := ψ : h(P~ ) ∈ C1(S\B ) , F~ ∈ C∞( 3 \ 0 ; 3) , (III.59) h,F~ 0 rα 0 R C where out/in i A~[F~t,t]−A~[F~t,t] ψ ~ := s − lim e ψh,κ(t) , (III.60) h,F t→+/−∞ and ~ ~ A[Ft, t] (III.61) is the L.S.Z. photon ﬁeld smeared out with the vector test function Z 3 ~ X d k ∗ ˆλ ~ −i|k|t+i~k·~y Ft(~y) := ~ε F (k) e (III.62) 3 p ~k,λ λ=± (2π) 2 |k| with ˆ X F~ (~k) := ~ε ∗ Fˆλ(~k) ∈ C∞( 3\{0} ; 3) . (III.63) ~k,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. INFRAPARTICLE STATES IN QED I. 25

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 (III.64) and (III.66), below, whose existence is established in Section VI.2.

∞ 3 iHt ~x −iHt Theorem III.3. Functions f ∈ C0 (R ), of the variable e t e , have strong limits, as t → ±∞, as operators acting on Hout/in, ~x s − lim eiHt f( ) e−iHtψout/in =: ψout/in (III.64) h,F~ f h , F~ t→+/−∞ t ∇~ E

~ σ where f~ := limσ→0 f(∇E ~ ). ∇EP~ P

Theorem III.4. The LSZ Weyl operators

~ ~ ~ ~ i A[Gt,t]−A[Gt,t] λ 2 3 −1 3 e : Gˆ (~k) ∈ L (R , (1 + |~k| )d k) , λ = ± , (III.65) have strong limits in Hout/in; i.e.,

~ ~ ~ ~ Wout/in(G~ ) := s − lim ei A[Gt,t]−A[Gt,t] . (III.66) t→+/−∞ exists. The limiting operators are unitary and satisfy the following properties: i) ~ ~ 0 out/in out/in 0 out/in 0 − ρ(G,G ) W (G~ ) W (G~ ) = W (G~ + G~ ) e 2 , (III.67)

where X Z λ ρ(G,~ G~ 0) = 2i Im Gˆλ(~k)Gˆ0 (~k) d3k . (III.68) λ ii) The mapping R 3 s −→ Wout/in(s G~ ) deﬁnes a strongly continuous one-parameter group of unitary operators. iii) iHτ out/in ~ −iHτ out/in ~ e W (G) e = W (G−τ ) (III.69) ~ where G−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 collective degrees of freedom are involved. The appearance of collective degrees of freedom is reﬂected in the representation of the asymptotic 26 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

out/in electromagnetic algebra Aph , which is non-Fock (but locally Fock; see Section VI.2). We will show that

Z kG~ k2 ~ out/in out out/in − 2 %∇~ E (G) 2 3 ~ 2 P~ ~ ψh,κ , W (G)ψh,κ = e e | h(P ) | d P, (III.70) where Z ~ ~ ~ 2 3 1/2 k Gk2 = |G(k)| d k , (III.71) and Z ~u · ~ε ∗ 1 X ~k,λ ~ 2 ˆλ ~ 3 %~u(G) := 2iRe α G (k) 3 d k . (III.72) ~ 2 ˆ λ Bκ |k| (1 − ~u · k) 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. 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 prop- agating along the light cone.

Remark: We point out that, in our deﬁnition of scattering states, we can directly accom- modate 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) A [F ] ... A [F ]ψh,κ := s − lim A[Ft , t] ... A[Ft , t]ψh,κ(t) (III.73) t→+∞ out ~ (m) ~ (1) which represents the state ψh,κ plus m asymptotic photons with wave functions F ,..., F , respectively. Analogously, we deﬁne

~in ~ (m0) ~in ~ (1) in ~ ~ (m0) ~ ~ (1) A [G ] ... A [G ]ψh,κ := s − lim A[−Gt , t] ... A[−Gt , t]ψh,κ(t) (III.74) t→−∞ This is possible because, apart from some higher order estimates to control the commutator i[H , ~x], and the photon creation operators in (III.73) (see for example [17]), we use the propagation estimate (III.43), 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. INFRAPARTICLE STATES IN QED I. 27

In particular, for any m, m0 ∈ N, we can define the S-matrix element 0 0 m,m ~ ~ ~out ~ (m) ~out ~ (1) out ~in ~ (m ) ~in ~ (1) in Sα ( {Fi} , {Gj} ) = A [F ] ... A [F ]ψh,κ , A [G ] ... A [G ]ψh,κ (III.75) 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, and an outgoing electron with wave function hout and soft photon energy threshold κout, respectively. m,m0 ~ ~ The expansion of Sα ( {Fi} , {Gj} ) in the finestructure constant α can be carried out, at least to leading order, along the lines of [1]. 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 Li´enard- Wiechert electromagnetic ﬁeld is observed. The precise mathematical statement is nD Z E lim lim |d~|2 ψout/in , eiHt d3y F (0, ~y) δ˜(~y − ~x − d~) e−iHtψout/in |d~|→∞ t→±∞ h,F~ µν h,F~ Z ~ o ∇EP~ 2 3 − Fµν (0, d~)|h(P~ )| d P = 0 , (III.76) where δ˜ is a smooth delta function, ~x is the electron position,

Fµν = ∂µAν − ∂νAµ (III.77) with π1/2α1/2 A (0, ~y) := − (III.78) 0 |~y − ~x| Z 3 X d k ~ ~ A (0, ~y) := − (~ ) e−ik·~ya∗ + (~∗ ) eik·~ya , (III.79) i q ~k,λ i ~k,λ ~k,λ i ~k,λ λ |~k|

~ ∇EP~ and Fµν is the electromagnetic ﬁeld tensor corresponding to a Li´enard-Wiechert solution for the current

1 1 3/2 2 (3) ~ 3/2 2 ~ (3) ~ ~ Jµ(t, ~y) := − 4π α δ (~y − ∇EP~ t) , 4π α ∇EP~ δ (~y − ∇EP~ t) , |∇EP~ | 1 ; (III.80) see the discussion in Section I. 28 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

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(P )| 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(P~ )|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 Section V, 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 (III.26)); • the rate t− of refinement of the cell partitions G (t).

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 (IV.1) by

σt σt σt σt D iγσ (~vl,∇E ,t) −iE t (t) iγσ (~vj ,∇E ,t) −iE t (t) E M (t) := e t P~ e P~ ψ , W (t)e t P~ e P~ ψ (IV.2) l,j l,σt σt,l,j j,σt where we use the notation Z 1 ∗ −i|~k|t ∗ i|~k|t 3 W (t) := exp α 2 ~η (~k) · ~ε a e − ~ε a e d k , (IV.3) σt,l,j l,j ~k,λ ~k,λ ~k,λ ~k,λ Bκ\Bσt and 1 ~vj 1 ~vl ~η (~k) := α 2 − α 2 . (IV.4) l,j ~ 3 ~ 3 |k| 2 (1 − bk · ~vj) |k| 2 (1 − bk · ~vl) We study the limit t → +∞; the case t → −∞ is analogous.

A. Asymptotic orthogonality

We introduce a family of operators Z µ 1 ∗ −i|~k|s ∗ i|~k|s 3 W (s) := exp µ α 2 ~η (~k) ·{~ε a e − ~ε a e }d k (IV.5) σt,l,j l,j ~k,λ ~k,λ ~k,λ ~k,λ Bκ\Bσt depending on a parameter µ ∈ R, and deﬁne

σt σt σt σt µ D iγσ (~vl,∇E ,s) −iE s (t) µ iγσ (~vj ,∇E ,s) −iE s (t) E M (t, s) := e t P~ e P~ ψ , W (s)e t P~ e P~ ψ (IV.6) cl,j l,σt σt,l,j j,σt

µ=1 for s ≥ t( 1). Obviously, Mcl,j (t, t) = Ml,j(t). INFRAPARTICLE STATES IN QED I. 29

The phase factor γ (~v , ∇Eσt , s) is chosen as follows: σt j P~ Z s Z γ (~v , ∇Eσt , s) := −α ∇~ Eσt · Σ~ (~k) cos(~k · ∇~ Eσt τ − |~k|τ)d3kdτ (IV.7) σt j P~ P~ ~vj P~ 1 B S \Bσ στ t −θ for s ≥ σt, and

− 1 θ Z σt Z γ (~v , ∇Eσt , s) := − α ∇~ Eσt · Σ~ (~k) cos(~k · ∇~ Eσt τ − |~k|τ)d3kdτ , (IV.8) σt j P~ P~ ~vj P~ 1 B S \Bσ στ t −θ for s < σt. As a function of µ, the scalar product in (IV.6) satisﬁes the ordinary diﬀerential equation µ dMcl,j(t, s) µ µ = − µ Cl,j,σt Mc (t, s) + r (t, s) , (IV.9) dµ l,j σt where Z ~ 2 3 Cl,j,σt := |~ηl,j(k)| d k , (IV.10) Bκ\Bσt and

~ σt σt µ D iγσ (~vl,∇E ,s) −iE s (t) r (t, s) := − e t P~ e P~ ψ , σt l,σt

σt σt µ iγσ (~vj ,∇~ E ,s) −iE s (t) E W (s)~a (~η )(s) e t P~ e P~ ψ σt,l,j σt l,j j,σt

~ σt σt D iγσ (~vl,∇E ,s) −iE s (t) + ~a (~η )(s)e t P~ e P~ ψ , σt l,j l,σt

σt σt µ iγσ (~vj ,∇~ E ,s) −iE s (t) E W (s)e t P~ e P~ ψ , σt,l,j j,σt with Z X ~ ~a (~η )(s) := ~η (~k) · ~ε ∗ a ei|k|s d3k . (IV.11) σt l,j l,j ~k,λ ~k,λ λ Bκ\Bσt The solution of the ODE (IV.9) is given by

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

0 Mcl,j(t, s) = 0 , (IV.13) since the supports in P~ of the two vectors ψ(t) , ψ(t) are disjoint (up to sets of measure 0), l,σt j,σt for arbitrary t and s. Furthermore, the vectors ψ(t) , ψ(t) are vacua for the annihilation part of the asymp- l,σt j,σt totic photon ﬁeld 1 under the dynamics generated by the Hamiltonian Hσt . This fact is

1The existence of the asymptotic field operator for a fixed cut-off dynamics is derived as explained in part B. of this section. 30 T.CHEN, J.FROHLICH,¨ AND A. PIZZO implied by condition (I 4) in Theorem III.1. As a consequence, we find that

µ lim rσ (t, s) = 0 , (IV.14) s→+∞ t for ﬁxed µ and t. Therefore, by dominated convergence, it follows that

1 lim Mcl,j(t, s) = 0 . (IV.15) s→+∞

1 Since Mcl,j(t, t) ≡ Ml,j(t), we have Z +∞ d 1 |Ml,j(t)| ≤ Mcl,j(t, s) ds . (IV.16) t ds d 1 To estimate | ds Mcl,j(t, s)| (see (IV.6)), we proceed as follows. Since we are interested in the limit t → ∞, and the integration domain on the r.h.s. of (IV.16) is [t, ∞), our aim is to show that

σ σt σt d iH t s iγσ (~vj ,∇E ,s) −iE s (t) e W (~v , s)e t P~ e P~ ψ (IV.17) ds σt j j,σt is integrable in s, and that the rate at which the time integral in (IV.16) converges to zero oﬀsets the growth of the number of cells in the partition. This allows us to conclude that X Ml,j(t) −→ 0 (IV.18) l,j (l6=j) in the limit t → +∞, and, as a corollary, N(t) Z X ~ 2 3 lim Ml,j(t) = |h(P )| d P, (IV.19) t→+∞ l,j as asserted in Theorem III.2. The convergence (IV.18) follows from the following theorem.

Theorem IV.1. The oﬀ-diagonal terms Ml,j(t), l 6= j, satisfy 1 | M (t) | ≤ C | ln σ |2 t−3 , (IV.20) l,j tη t for some constants C < ∞ and η > 0, where C is independent of l and j, and η > 0 is independent of . In particular, Ml,j(t) → 0, as t → +∞.

As a corollary, we ﬁnd X 1 1 | M (t) | ≤ C0 N 2(t) | ln σ |2 t−3 ≤ C | ln σ |2 t3 (IV.21) l,j tη t tη t 1≤l6=j≤N(t) 3 η since N(t) ≈ t . We conclude that, for < 4 , (IV.18) follows. INFRAPARTICLE STATES IN QED I. 31

B. Time derivative and infrared tail

We now proceed to prove Theorem IV.1. 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 Section V. To control (IV.16), we focus on the derivative

σ σt σt d iH t s iγσ (~vj ,∇~ E ,s) −iE s (t) e W (~v , s) e t P~ e P~ ψ (IV.22) ds σt j j,σt σ σt σt iH t s σ iγσ (~vj ,∇~ E ,s) −iE s (t) t t P~ P~ = i e [HI , Wσt (~vj, s)] e e ψj,σt (IV.23)

σ ~ ~ σt σt iH t s dγσt (~vj, ∇EP~ , s) iγσ (~vj ,∇E ,s) −iE s (t) + i e W (~v , s) e t P~ e P~ ψ , (IV.24) σt j ds j,σt where ~ ~ σ 1 Aσt (~x) · Aσt (~x) H t := α 2 ~p · A~ (~x) + α . (IV.25) I σt 2 −isH0 isH0 σt σt We have used that Wσt (~vj, s) = e Wσt (~vj, 0)e , where H0 := H − HI is the free Hamiltonian, to obtain the commutator in (IV.23). We rewrite the latter in the form σt ∗ σt σt [ HI , Wσt (~vj, s) ] = Wσt (~vj, s) Wσt (~vj, s) HI Wσt (~vj, s) − HI , (IV.26) and use that

1 Z ∗ ~ ~ 2 ~ ~ ~ ~ 3 Wσt (~vj, s) Aσt (~x) Wσt (~vj, s) = Aσt (~x) + α Σ~vj (k) cos(k · ~x − |k|s) d k (IV.27) Bκ\Bσt (see (III.8)), where Σl (~k) is deﬁned in (III.27). We can then write the term (IV.23) as ~vj

σ Z iH t s σt 3 ~ ~ (IV.23) = i e Wσt (~vj, s)α i[H , ~x] · d k Σ~vj (k) × (IV.28) Bκ\Bσt σt σt −iE s iγσ (~vj ,∇~ E ,s) (t) ~ ~ P~ t P~ × cos(k · ~x − |k|s) e e ψj,σt 2 σ α Z + i eiH t sW (~v , s) Σ~ (~k) cos(~k · ~x − |~k|s)d3k · (IV.29) σt j 2 ~vj Bκ\Bσt

Z σt σt 3 iγσ (~vj ,∇~ E ,s) −iE s (t) ~ t P~ P~ · d q Σ~vj (~q) cos(~q · ~x − |~q|s) e e ψj,σt , Bκ\Bσt

1 σt 2 ~ where we recall that i[H , ~x] = ~p + α Aσt (~x); see (III.40). From the decay estimates provided by Lemma A.2 in the Appendix one concludes that the norm of (IV.29) is integrable in s, and that ∞ Z 1 3 2 − 2 ds k (IV.29) k ≤ η | ln σt | t (IV.30) t t for some η > 0 independent of . 32 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

The analysis of (IV.28) is more involved. Our argument will eventually involve the derivative of the phase factor in (IV.24). To begin with, we write (IV.28) as

σ s iH t σt 1 (IV.28) = i e Wσ (~vj, s)α i[H , ~x] · × (IV.31) t Hσt + i

Z σt σt σ 3 −iE s iγσ (~vj ,∇~ E ,s) (t) t ~ ~ ~ ~ P~ t P~ ×(H + i) d k Σ~vj (k) cos(k · ~x − |k|s) e e ψj,σt . Bκ\Bσt

Pulling the operator (Hσt + i) through to the right, the vector (IV.31) splits into the sum of a term involving the commutator [Hσt , ~x],

σ s iH t σt 1 i e Wσ (~vj, s)α i[H , ~x] · × (IV.32) t Hσt + i

Z σt σt 3 σ iγσ (~vj ,∇~ E ,s) −iE s (t) ~ ~ t ~ ~ t P~ P~ × d k Σ~vj (k)[ H , cos(k · ~x − |k|s)] e e ψj,σt , Bκ\Bσt and

σ Z iH t s σt 3 ~ ~ i e Wσt (~vj, s)α i[H , ~x] · d k Σ~vj (k)× (IV.33) Bκ\Bσt 1 iγ (~v ,∇~ Eσt ,s) −iEσt s (t) ~ ~ σt j ~ ~ σt × cos(k · ~x − |k|s)e P e P (E + i)ψj,σ . Hσt + i P~ t

To control (IV.32) and (IV.33), we invoke a propagation estimate for the electron position operator as follows. Due to condition (I 3) in Theorem III.1, we can introduce a C∞−function 3 χh(~y), ~y ∈ R , such that

• χ (∇~ Eσt ) ≡ 1 for P~ ∈ supp h, uniformly in σ . h P~ t

• χh(~y) = 0 for |~y| < νmin and |~y| > νmax .

It is shown in Theorem A.3 of the Appendix that, for θ < 1 suﬃciently close to 1 and s large, the propagation estimate

σt σt ~x iγσ (~vj ,∇~ E ,s) −iE s (t) χ ( )e t P~ e P~ ψ h s j,σt σt σt σ iγσ (~vj ,∇~ E ,s) −iE s (t) − χ (∇~ E t )e t P~ e P~ ψ (IV.34) h P~ j,σt 1 1 ≤ ν 3 | ln(σt)| , s t 2 holds, where ν > 0 is independent of . The argument uses the H¨older regularity of ∇~ Eσ and P~ of Φσ listed under properties ( 2) in Theorem III.1, diﬀerentiability of h(P~ ), and (III.17). P~ I INFRAPARTICLE STATES IN QED I. 33

We continue with the discussion of the expressions (IV.32) and (IV.33). We split (IV.33) into two parts using the deﬁnitions

1 Z κ σt ~ ~ ~ ~ 3 −θ J | S (s) := α i[H , ~x] · Σ~v (k) cos(k · ~x − |k|s)d k if s ≥ σt σs σt j H + i Bκ\B S σs −θ := (IV.33) if s < σt , (IV.35) and

S 1 Z σs σt ~ ~ ~ ~ 3 −θ J | (s) := α i[H , ~x] · Σ~v (k) cos(k · ~x − |k|s)d k if s ≥ σt σt σt j H + i B S \Bσ σs t −θ := 0 if s < σt , (IV.36)

S −θ where we refer to σs := s as the slow infrared cut-oﬀ. S σt To control J |σt (s) in (IV.36), we deﬁne the ”infrared tail”

~x iHσt s −iHσt s dγ (~v , , s) σ 1 d(e ~x (s)e ) σ bσt j s := α e−iH t s h eiH t s · (IV.37) ds Hσt + i ds Z · Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s) d3k if s−θ ≥ σ , ~vj P~ t B S \Bσ σs t −θ := 0 if s < σt

~x where ~xh(s) := ~xχh( s ). Summarizing, we can write (IV.22) as

(IV.22) = (IV.29) + σ σ σt t t iH s κ iγσt (~vj ,∇E ~ ,s) −iE ~ s σt (t) + i e Wσ (~vj, s) J | S (s) e P e P (E + i) ψ (IV.38) t σs P~ j,σt σ S σt σt iH t s σ iγσ (~vj ,∇E ,s) −iE s σ (t) + i e W (~v , s) J | s (s) e t P~ e P~ (E t + i) ψ (IV.39) σt j σt P~ j,σt σt σ dγσ (~vj, ∇E , s) σt σt iH t s t P~ iγσ (~vj ,∇E ,s) −iE s (t) + i e W (~v , s) e t P~ e P~ ψ (IV.40) σt j ds j,σt σ iH t s σt 1 + i e Wσ (~vj, s) α i [H , ~x] · × (IV.41) t Hσt + i

Z σt σt 3 σ iγσ (~vj ,∇~ E ,s) −iE s (t) ~ ~ t ~ ~ t P~ P~ × d k Σ~vj (k)[H , cos(k · ~x − |k|s)] e e ψj,σt , Bκ\Bσt where we recall that (IV.29) satisﬁes (IV.30). We claim that

Z ∞ 1 [(IV.38) + (IV.39) + (IV.40)] ds ≤ | ln σ |2 t−3 , (IV.42) η t t t 34 T.CHEN, J.FROHLICH,¨ AND A. PIZZO for some η > 0 depending on θ, but independent of . This is obtained from Z ∞

[(IV.38) + (IV.39) + (IV.40)] ds (IV.43) t Z ∞ 1 Z σt 3 ~ ~ ~ ~ ≤ ds α i[H , ~x] · d k Σ~v (k) cos(k · ~x − |k|s) × Hσt + i j t Bκ\Bσt σt σt h σ ~x i iγσ (~vj ,∇E ,s) −iE s σ (t) × χ (∇~ E t ) − χ ( ) e t P~ e P~ (E t + i) ψ (IV.44) h P~ h s P~ j,σt ∞ Z ~x σt σt κ iγσt (~vj ,∇E ~ ,s) −iE ~ s σt (t) + ds J | S (s)χh( )e P e P (E + i) ψ (IV.45) σs P~ j,σt t s Z ∞ ~x σt h S ~x dγσt (~vj, , s) i + ds eiH s W (~v , s) J |σs (s)χ ( ) − b s × (IV.46) σt j σt h t s ds σt σt iγσ (~vj ,∇~ E ,s) −iE s σ (t) × e t P~ e P~ (E t + i) ψ P~ j,σt ∞ ~ σt ~x Z σ hdγσt (~vj, ∇E , s) dγ (~v , , s)i + ds eiH t s W (~v , s) P~ − bσt j s × (IV.47) σt j t ds ds σt σt iγσ (~vj ,∇~ E ,s) −iE s σ (t) × e t P~ e P~ (E t + i) ψ P~ j,σt using the following arguments:

1 2 −3 • The term (IV.44) can be bounded from above by tη | ln σt| t , for some η > 0 independent of , due to the propagation estimate for (III.43) and Lemma A.2, which show that the integrand has a suﬃciently strong decay in s. S ~x • In (IV.45), the slow cut-oﬀ σt and the function χh( s ) make the norm integrable in s with the desired rate, for a suitable choice of θ < 1. In particular, we can exploit that Z ~x sθ 3 ~ ~ ~ ~ sup d k Σ(k,~vj) cos(k · ~x − |k|s) χh( ) ≤ O( 2 ) , (IV.48) 3 ~x∈ Bκ\B S s s R σs see Lemma A.2 in the Appendix. • In (IV.46), only terms integrable in s and decaying fast enough to satisfy the bound (IV.34) are left after subtracting dγ (~v , ~x , s) bσt j s (IV.49) ds S σs from J |σt (s). This is explained in detail in the proof of Theorem A.4 in the Appendix. • To bound (IV.47), 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. We note that due to the vector interaction in non-relativistic QED, this argument is more complicated here than in the Nel- son model treated in [24] where the interaction term is scalar. Here, we have to INFRAPARTICLE STATES IN QED I. 35

show (see Theorem A.4) that in the integral with respect to s, the pointwise velocity σ σ eiH t si[~x, Hσt ]e−iH t s can be replaced by the mean velocity ∇~ Eσt at asymptotic times. P~ Finally, to control (IV.41), we observe that the commutator introduces additional decay ~x 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 κ arguments that were applied to J | S (s) in (IV.44), (IV.45). σs Combining the above arguments, the proof of Theorem IV.1 is completed. 36 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

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.

V.1. Three key steps. As anticipated in Section III.4, 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 (III.48) – (III.50):

I) The term

(t2) (t1) ∆ψ(t2, σt2 , G → G ) N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t2) −iE t2 (t1) 2 t2 P~ P~ = e Wσt (~vj, t2) e e ψ (V.2) 2 j,σt2 j=1 N(t ) 1 σ σ t2 t2 iHt X X iγσ (~v ,∇~ E ,t2) −iE t2 (t2) 2 t2 l(j) P~ P~ − e Wσt (~vl(j), t2)e e ψ , 2 l(j),σt2 j=1 l(j)

(t2) (t1) accounts for the change of the partition G → G in ψh,κ(t2), where l(j) labels

(t2) (t1) (t1) the sub-cells belonging to the sub-partition G ∩ Gj of Gj , and where we deﬁne

σ σ ~ t2 ~ t1 ~vl(j) ≡ ∇E ~ ∗ and ~vj ≡ ∇E ~ ∗ ; Pl(j) Pj

II) the term

(t1) ∆ψ(t2 → t1, σt2 , G ) N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t2 P~ P~ = e Wσt (~vj, t1) e e ψ (V.3) 2 j,σt2 j=1 N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t2) −iE t2 (t1) 2 t2 P~ P~ − e Wσt (~vj, t2) e e ψ , 2 j,σt2 j=1

accounts for the subsequent change of the time variable, t2 → t1, for the ﬁxed parti-

(t1) tion G , and the fixed infrared cut-off σt2 ; and finally, INFRAPARTICLE STATES IN QED I. 37

III) the term

(t1) ∆ψ(t1, σt2 → σt1 , G ) N(t ) 1 σ σ t1 t1 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t1 P~ P~ = e Wσt (~vj, t1)e e ψ (V.4) 1 j,σt1 j=1 N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t2 P~ P~ − e Wσt (~vj, t1)e e ψ 2 j,σt2 j=1

accounts for the change of the infrared cut-oﬀ, σt2 → σt1 . Our goal is to prove

2 ρ k ψh,κ(t2) − ψh,κ(t1) k ≤ O (ln(t2)) /t1 , (V.5) for some ρ > 0. To this end, we must take these three steps in the order displayed above. As a corollary of the bound (V.5), we obtain Theorem III.2 by telescoping. The arguments in our proof are very similar to those in [24], but a number of mod- iﬁcations are necessary because of the vector nature of the QED interaction. For these modiﬁcations, we provide detailed explanations.

V.2. Refining the cell partition. In this section, we discuss step (V.2) in which the momentum space cell partition is refined. It is possible to apply the methods developed in [24], up to some minor modifications. We will prove that

(t2) (t1) 2 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 IV.1. That is, we ﬁrst express ψ(t1) as j,σt2 Z Z (t ) σt X σt X (t ) ψ 1 = h(P~ )Ψ 2 d3P = h(P~ )Ψ 2 d3P = ψ 2 . (V.7) j,σt ~ ~ l(j),σt 2 (t1) P (t2) P 2 Gj l(j) Gj l(j) Then,

N(t ) 1 σ t2 X X D −iE t2 (t2) P~ (V.6) = [Wcσt (~vl(j), t2) − Wcσt (~vj, t2)]e ψ , 2 2 l(j),σt2 j,j0=1 l(j),l0(j0)

σ t2 −iE t2 (t2) E 0 0 0 P~ [Wcσt (~vl (j ), t2) − Wcσt (~vj , t2)]e ψ 0 0 , (V.8) 2 2 l (j ),σt2 38 T.CHEN, J.FROHLICH,¨ AND A. PIZZO where we deﬁne

σ t2 iγσ (~vj ,∇~ E ,t2) W (~v , t ) := W (~v , t ) e t2 P~ , (V.9) cσt2 j 2 σt2 j 2 σ t2 iγσ (~v ,∇~ E ,t2) W (~v , t ) := W (~v , t ) e t2 l(j) P~ . (V.10) cσt2 l(j) 2 σt2 l(j) 2 Following the analysis in Section IV.1, one ﬁnds that the sum over pairs (l0(j0) , l(j)) with 0 0 − η either l 6= l or j 6= j can be bounded by O(t2 ), provided that < 4 , as in (IV.21). Let h i stand for the expectation value with respect to the vector Ψ. Then, we are Ψe e left with the diagonal terms

N(t1) X X D E 2 − W∗ (~v , t ) W (~v , t ) − W∗ (~v , t )W (~v , t ) , (V.11) cσt l(j) 2 cσt2 j 2 cσt j 2 cσt2 l(j) 2 2 2 Ψe j=1 l(j)

σ t2 −iE t2 (t2) labeled by pairs (l(j) , l(j)), where Ψe ≡ e P~ ψ in the case considered here. For l(j),σt2 each term ∗ Wc (~vl(j), t2)Wcσt (~vj, t2) , (V.12) σt2 2 Ψe we can again invoke the arguments developed for oﬀ-diagonal elements indexed by (l, j) (where l 6= j) from Section IV.1.

In particular, we deﬁne for s > t2

D σt σt E µ −iE ~ s (t2) ∗ µ µ −iE ~ s (t2) M (t2, s) := e P ψ , W (~vj, s) W (~vl(j), s) e P ψ , (V.13) c[l(j),~vj ],[l(j),~vl(j)] l(j),σt cσt2 cσt2 l(j),σt where

σ σ ~ t2 ~ t2 ∗ µ µ −iγσt (~vj ,∇E ~ ,s) µ ∗ µ iγσt (~vl(j),∇E ~ ,s) W (~vj, s) W (~vl(j), s) = e 2 P W (~vj, s) W (~vl(j), s) e 2 P , cσt2 cσt2 σt2 σt2 (V.14) and Z µ ∗ µ 1 ~ ∗ −i|~k|s ∗ i|~k|s 3 W (~vj, s) W (~vl(j), s) = exp µ α 2 ~ηj,l(j)(k) · ~ε~ a e − ~ε a~ e d k , σt2 σt2 k,λ ~k,λ ~k,λ k,λ Bκ\Bσt (V.15) with µ a real parameter.

Proceeding similarly as in (IV.12), the solution of the ODE analogous to (IV.9) for µ M (t2, s) at µ = 1 consists of a contribution at µ = 0, which remains non-zero c[l(j),~vj ],[l(j),~vl(j)] as s → ∞, and a remainder term that vanishes in the limit s → ∞. In fact,

1 lim Mc[l(j),~v ],[l(j),~v ](t2, s) (V.16) s→+∞ j l(j)

C σ σ − 1 σ − 1 j,l(j) t2 t2 θ t2 θ − D iγσ (~vj ,∇~ E ,σ ) (t2) iγσ (~v ,∇~ E ,σ ) (t2) E = e 2 e t2 P~ t2 ψ , e t2 l(j) P~ t2 ψ , l(j),σt2 l(j),σt2 INFRAPARTICLE STATES IN QED I. 39 where Z C := |~η (~k)|2d3k , (V.17) j,l(j),σt2 j,l(j) Bκ\Bσ t2 ~ as in (IV.10), with ~ηj,l(j)(k) deﬁned in (IV.4). Hence, (V.11) is given by the sum of

N(t2) Z ∞ X X d 1 − Mc (t2, s) ds ds [`(j),~vj ],[`(j),~v`(j)] j=1 `(j) t2 N(t2) Z ∞ X X d 1 − Mc (t2, s) ds (V.18) ds [`(j),~v`(j)],[`(j),~vj ] j=1 `(j) t2 and

N(t2) C l(j),j,σt X X D (t2) σt 2 (t2) E ~ 2 − 2 ψ , 2 − 2 cos ∆γσt (~vj − ~vl(j), ∇E , t2) e ψ , (V.19) l(j),σt2 2 P~ l(j),σt2 j=1 `(j) where

σ σ − 1 σ − 1 ~ t2 ~ t2 θ ~ t2 θ ∆γσ (~vj − ~vl(j), ∇E , t2) := γσ (~vl(j), ∇E , σ ) − γσ (~vj, ∇E , σ ) . (V.20) t2 P~ t2 P~ t2 t2 P~ t2 The arguments that have culminated in Theorem IV.1 also imply that the sum (V.18) can −4 be bounded by O(t2 ), for η > 4. The leading contribution in (V.6) is represented by the sum (V.19) of diagonal terms (with respect to G (t2)), which can now be bounded from above. It suﬃces to show that

Cl(j),j,σ σ t2 1 ~ t2 − sup 2 − 2 cos ∆γσ (~vj − ~vl(j), ∇E , t2) e 2 ≤ log t2 (V.21) t2 P~ η P~ ∈S t1 for some η > 0 that depends on . To see this, we note that the lower integration bound in the integral (V.17) contributes a factor to (V.21) proportional to log t2. In Lemma A.1, it is proven that

σ 1 σ 1 ~ t2 − ~ t2 − γσ (~vj, ∇E , (σt ) θ ) − γσ (~vl(j), ∇E , (σt ) θ ) ≤ O(|~vj − ~vl(j)|) . (V.22) t2 P~ 2 t2 P~ 2 σ σ ~ t1 ~ t2 ~ We can estimate the diﬀerence ~vj − ~vl(j) = ∇E ~ ∗ − ∇E ~ ∗ , which also appears in ~ηl(j),j(k), Pj Pl(j) using condition (I 2) of Theorem III.1. This yields the -dependent negative power of t1 in (V.21).

V.3. Shifting the time variable for a ﬁxed cell partition and infrared cut-oﬀ. In this subsection, we prove that

(t1) 2 2 ρ ∆ψ(t2 → t1, σt2 , G ) ≤ O (ln(t2)) /t1 (V.23) 40 T.CHEN, J.FROHLICH,¨ AND A. PIZZO for some ρ > 0; see (V.3). 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 (IV.22).

For t1 ≤ s ≤ t2, we deﬁne s σ Z σ Z σ ~ t2 ~ t2 ~ ~ ~ ~ t2 ~ 3 γσ (~vj, ∇E , s) := − ∇E · Σ~v (k) cos(k · ∇E τ − |k|τ) d k dτ . (V.24) t2 P~ P~ j P~ 1 B S \Bσ στ t2 Then, we estimate

t2 σ σ Z σt t2 t2 d iH 2 s iγσ (~vj ,∇~ E ,s) −iE s (t1) t2 P~ P~ ds e Wσt (~vj, 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 (IV.22), see also the remark after Theorem A.3, by deﬁning a tail in a similar fashion. The only modiﬁcation to be added is that, apart from two terms analogous to (IV.23), (IV.24), we now also have to consider σ σ t2 t2 iHt σ iγσ (~vj ,∇~ E ,s) −iE s (t1) 2 t2 t2 P~ P~ i e (H − H ) Wσt (~vj, s)e e ψ , (V.26) 2 j,σt2 which enters from the derivative in s of the operator underlined in σ σ σt σt t2 t2 iHs −iH 2 s iH 2 s iγσ (~vj ,∇~ E ,s) −iE s (t1) t2 P~ P~ e e e Wσt (~vj, s)e e ψ . (V.27) 2 j,σt2 To control the norm of (V.26), we observe that ~ ~ 1 A<σt · A<σt σt2 ~ 2 2 H − H = α 2 i[H, ~x] · A<σ − α , (V.28) t2 2 where Z 3 X d k ∗ ∗ A~ := ~ε~ b + ~ε b~ , (V.29) <σt2 q k,λ ~k,λ ~k,λ k,λ Bσ λ=± t2 |~k| and we note that [H, ~x] · A~ = A~ · [H, ~x] , <σt2 <σt2 because of the Coulomb gauge condition. Moreover,

∗ ~ W (~vj, s) i[H, ~x] Wσ (~vj, s) = i[H, ~x] + hs(~x) (V.30) σt2 t2 ~ with khs(~x)k < O(1) and [b , ~h (~x)] = [~h (~x) , A~ ] = 0 . ~k,λ s s <σt2 Furthermore, we have

(t1) b~ ψ = 0 k,λ j,σt2 for ~k ∈ B , and σt2 ~ (t1) ~ ~ (t1) kA<σt ψ k , kA<σt · A<σt ψ k ≤ O(σt ) . (V.31) 2 j,σt2 2 2 j,σt2 2 INFRAPARTICLE STATES IN QED I. 41

The estimate

~ (t1) (t1) (t1) kA<σt · [H, ~x] ψ k ≤ O( σt k[H, ~x] ψ k + kψ k ) , (V.32) 2 j,σt2 2 j,σt2 j,σt2 holds, where − 3 k[H, ~x]ψ(t1) k ≤ O(t 2 ) , (V.33) j,σt2 1 because

(t1) σt (t1) k[H, ~x]ψ k ≤ c1k(H 2 + i)ψ k j,σt2 j,σt2 for some constant c1, and − 3 kψ(t1) k = O(t 2 ) . j,σt2 1 Consequently, we obtain that

σ σ t2 t2 σ iγσ (~vj ,∇~ E ,s) −iE s (t1) t2 t2 P~ P~ (H − H ) Wσt (~vj, s)e e ψ 2 j,σt2 (t1) (t1) −3/2 ≤ O( σt k[H, ~x] ψ k + kψ k ) ≤ O( σt t ) . (V.34) 2 j,σt2 j,σt2 2 Following the procedure in Section II.2.1, B., one can also check that

σ σ σt σt t2 t2 iHt −iH 2 s d iH 2 s iγσ (~vj ,∇~ E ,s) −iE s (t1) 2 t2 P~ P~ e e e Wσt (~vj, s)e e ψj,σ ds 2 t2 3 1 2 − 2 ≤ O( η | ln σt2 | t1 ) , (V.35) t1 η for some η > 0. Similarly as in (IV.21), we choose small enough that 4 > . (t1) 3 The number of cells in the partition G is N(t1) ≈ t1 . Therefore, summing over all cells, we get 3 3 − 2 1 2 − 2 O N(t1) t1 σt2 + O N(t1) η | ln σt2 | t1 , (V.36) t1 as an upper bound on the norm of the term in (V.3). −β The parameter β in the deﬁnition of σt2 = t2 can be chosen arbitrarily large, independently of . Hereby, we arrive at the upper bound claimed in (V.5).

V.4. Shifting the infrared cut-oﬀ. In this section, we prove that

(t1) 2 2 ρ k ∆ψ(t1, σt2 → σt1 , G ) k ≤ O (ln(t2)) /t1 (V.37) for some ρ > 0; see (V.4). 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 (V.4),

N(t ) 1 σ σ t2 t2 iHt X iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 t2 P~ P~ e Wσt (~vj, t1) e e ψ , (V.38) 2 j,σt2 j=1 42 T.CHEN, J.FROHLICH,¨ AND A. PIZZO as N(t ) 1 σ σ t2 t2 iHt X ∗ σt σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 ~ 2 t2 P~ P~ e Wσt (~vj, t1) W (∇E ) Wσt (∇E ) e e ψ , (V.39) 2 σt2 P~ 2 P~ j,σt2 j=1 and to group the terms appearing in (V.39) in such a way that, cell by cell, we consider the new dressing operator

σt σt −iE 2 t iHt1 ∗ ~ 2 ~ 1 e Wσ (~vj, t1) W (∇E ) e P , (V.40) t2 σt2 P~ which acts on Z σ (t1) ~ t2 3 Φj,σt := h(P )Φ ~ d P, (V.41) 2 (t1) P Gj σ ~ σ σ (t1) where Φ = Wσ(∇E )Ψ , see (III.22). The key advantage is that the vector Φ inherits P~ P~ P~ j,σt2 the H¨olderregularity of Φσ ; see (III.13) in condition ( 2) of Theorem III.1. We will refer P~ I to (V.41) as an infrared-regular vector. Accordingly, (V.39) now reads

N(t ) 1 σ σ t2 t2 iHt X ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t2 P~ P~ e Wσt (~vj, t1) W (∇E ) e e Φ , (V.42) 2 σt2 P~ j,σt2 j=1 and we proceed as follows.

A. Shifting the IR cutoﬀ in the infrared-regular vector

First, we substitute N(t ) 1 σ σ t2 t2 iHt X ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t2 P~ P~ e Wσt (~vj, t1)W (∇E ) e e Φ (V.43) 2 σt2 P~ j,σt2 j=1 N(t ) 1 σ σ t1 t1 iHt X ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t1 P~ P~ −→ e Wσt (~vj, t1)W (∇E ) e e Φ , 2 σt2 P~ j,σt1 j=1 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 (V.37). The necessary ingredients are: 1) Condition (I 1) in Theorem III.1. 2) The estimate

σ σ 1 (1−δ) ~ t2 ~ t1 2 1−θ |γσ (~vj, ∇E , t1) − γσ (~vj, ∇E , t1)| ≤ O σ t + O t1 σt , t2 P~ t1 P~ t1 1 1

for t2 > t1 1, proven in Lemma A.1. The parameter 0 < θ < 1 is the same as the one in (III.25).

(t1) 3) The cell partition G depends on t1 < t2. INFRAPARTICLE STATES IN QED I. 43

4) The parameter β can be chosen arbitrarily large, independently of , so that the −β 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 (V.43) 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 (V.43) is bounded from above by

1 3 2 (1−δ) − 2 O( t1 σt1 t1 ) , (V.44) − 3 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 3 2 (1−δ) 1− 2 O( N(t1) σt1 t1 ) (V.45)

3 where N(t1) ≈ t1 , by 3). Picking β sufficiently large, by 4), we find that the norm difference −η of the two vectors in (V.43) is bounded by t1 , for some η > 0. This agrees with the bound stated in (V.37).

B. Shifting the IR cutoﬀ in the dressing operator

Subsequently to (V.43), we substitute N(t ) 1 σ σ t1 t2 iHt X ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t1 P~ P~ e Wσt (~vj, t1)W (∇E )e e Φ (V.46) 2 σt2 P~ j,σt1 j=1 N(t ) 1 σ σ t1 t1 iHt X ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 1 t1 P~ P~ −→ e Wσt (~vj, t1)W (∇E )e e Φ , 1 σt1 P~ j,σt1 j=1 where σt2 → σt1 in the underlined operators. A crucial point in our argument is that when σ σ (> σ ) tends to 0, the H¨oldercontinuity of Φ t1 in P~ oﬀsets the (logarithmic) divergence t1 t2 P~ in t2 which arises from the dressing operator.

We subdivide the shift σt2 → σt1 in σ σ ∗ ~ t2 ∗ ~ t1 Wσ (~vj, t1)W (∇E ) −→ Wσ (~vj, t1)W (∇E ) (V.47) t2 σt2 P~ t1 σt1 P~ into the following three intermediate steps, where the operators modiﬁed in each step are underlined: Step a)

σ ∗ ∗ ~ t2 Wσ (~vj, t1)W (~vj)Wσ (~vj)W (∇E ) (V.48) t2 σt2 t2 σt2 P~ σ ∗ ∗ ~ t2 −→ Wσ (~vj, t1)W (~vj)Wσ (~vj)W (∇E ) , t1 σt1 t2 σt2 P~ 44 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

Step b)

σ ∗ ∗ ~ t2 Wσ (~vj, t1)W (~vj)Wσ (~vj)W (∇E ) (V.49) t1 σt1 t2 σt2 P~ σ ∗ ∗ ~ t1 −→ Wσ (~vj, t1)W (~vj)Wσ (~vj)W (∇E ) , t1 σt1 t2 σt2 P~

Step c)

σ ∗ ∗ ~ t1 Wσ (~vj, t1)W (~vj)Wσ (~vj)W (∇E ) (V.50) t1 σt1 t2 σt2 P~ σ ∗ ∗ ~ t1 −→ Wσ (~vj, t1)W (~vj)Wσ (~vj)W (∇E ) . t1 σt1 t1 σt1 P~

Analysis of Step a)

In step a), we analyze the diﬀerence between the vectors

σ σ t1 t2 iHt ∗ ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t1 P~ P~ e Wσt (~vj, t1)W (~vj)Wσt (~vj)W (∇E )e e Φ (V.51) 2 σt2 2 σt2 P~ j,σt1 and

σ σ t1 t2 iHt ∗ ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t1 P~ P~ e Wσt (~vj, t1)W (~vj)Wσt (~vj)W (∇E )e e Φ , (V.52) 1 σt1 2 σt2 P~ j,σt1 for each cell in G (t1). Our goal is to prove that

k (V.51) − (V.52) k ≤ const log t2 P (t1, t2) , (V.53) where

~ ~ σ −i(|k|t1−k·~x) ∗ ~ t2 P (t1, t2) := sup (e − 1)Wσ (~vj)W (∇E )) × (V.54) t2 σt2 P~ ~k∈Bσ t1 σ σ ~ t2 t2 iγσt (~vj ,∇E ,t1) −iE t1 (t1) −η × e 2 P~ e P~ Φ ≤ O( t log t2 ) j,σt1 1 as t1 → +∞, for some η > 0, and for β large enough. Using the identity

∗ ∗ Wσ (~vj, t1)W (~vj) = Wσ (~vj, t1)W (~vj) × t2 σt2 t1 σt1 iα Z ~v · Σ~ (~k) sin(~k · ~x − |~k|t ) × exp d3k j ~vj 1 × (V.55) ~ 2 Bσ \Bσ |k|(1 − k · ~v ) t1 t2 b j ~ ~ Z 3 ~v ·{~ε b∗ (e−i(|k|t1−k·~x) − 1) − h.c. } 1 d k j ~k,λ ~k,λ × exp α 2 , q ~ Bσ \Bσ |k|(1 − k · ~v ) t1 t2 |~k| b j INFRAPARTICLE STATES IN QED I. 45 the diﬀerence between (V.51) and (V.52) is given by

iα Z ~v · Σ~ (~k) sin(~k · ~x − |~k|t ) iHt1 ∗ 3 j ~vj 1 e Wσt (~vj, t1)Wσ (~vj) exp d k × 1 t1 ~ 2 Bσ \Bσ |k|(1 − k · ~v ) t1 t2 b j ~ ~ Z 3 ~v ·{~ε b∗ (e−i(|k|t1−k·~x) − 1) − h.c. } h 1 d k j ~k,λ ~k,λ i × exp α 2 − I × q ~ Bσ \Bσ |k|(1 − k · ~v ) t1 t2 |~k| b j σ σ t1 t2 ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) ~ 2 t1 P~ P~ × Wσt (~vj)W (∇E ) e e Φ (V.56) 2 σt2 P~ j,σt1 h iα Z ~v · Σ~ (~k) sin(~k · ~x − |~k|t ) i iHt1 ∗ 3 j ~vj 1 + e Wσt (~vj, t1)Wσ (~vj) exp d k − I × 1 t1 ~ 2 Bσ \Bσ |k|(1 − k · ~v ) t1 t2 b j σ σ t1 t2 ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) ~ 2 t1 P~ P~ × Wσt (~vj)W (∇E ) e e Φ , (V.57) 2 σt2 P~ j,σt1 where I is the identity operator in H. The norm of the vector (V.56) equals

~ ~ Z 3 ~v ·{~ε b∗ (e−i(|k|t1−k·~x) − I) − h.c. } h 1 d k j ~k,λ ~k,λ i exp α 2 − I × q ~ Bσ \Bσ |k|(1 − k · ~v ) t1 t2 |~k| b j σ σ t1 t2 ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) ~ 2 t1 P~ P~ × Wσt (~vj)W (∇E )e e Φ . (V.58) 2 σt2 P~ j,σt1 We now observe that • for ~k ∈ B , σt1

σ ∗ ~ t2 b~ Wσ (~vj)W (∇E ) (V.59) k,λ t2 σt2 P~ σ ∗ ~ t2 = Wσ (~vj)W (∇E ) b~ (V.60) t2 σt2 P~ k,λ σ ∗ ~ t2 ~ + Wσ (~vj)W (∇E ) f~ (~vj, P ) , (V.61) t2 σt2 P~ k,λ where Z 3 ~ 2 d k |f~k,λ(~vj, P )| ≤ O(ln σt2 ) (V.62) Bσ \Bσ t1 t2 ~ uniformly in ~vj, and in P ∈ S, and where j enumerates the cells. • for ~k ∈ B , σt1 σ σ ~ t2 t2 iγσt (~vj ,∇E ,t1) −iE t1 (t1) b~ 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

(V.58) ≤ c | log σt2 | P (t1, t2) , (V.64) 46 T.CHEN, J.FROHLICH,¨ AND A. PIZZO for some ﬁnite constant c as claimed in (V.53), where

σ σ t2 t2 −i(|~k|t −~k·~x) ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t2 P~ P~ P (t1, t2) = sup (e − 1) Wσt (~vj)W (∇E )e e Φ , 2 σt2 P~ j,σt1 ~k∈Bσ t1 (V.65) as deﬁned in (V.54). To estimate P (t1, t2), we regroup the terms inside the norm into

σ σ t2 t2 −i(|~k|t −~k·~x) ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 ~ 2 t2 P~ P~ (e − 1)Wσt (~vj) W (∇E ) e e Φ 2 σt2 P~ j,σt1 σ σ t2 t2 ∗ σt −i(|~k|t −~k·~x) iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) ~ 2 1 t2 P~ P~ = Wσt (~vj) W (∇E )(e − I) e e Φ (V.66) 2 σt2 P~ +~k j,σt1 σ σ t2 t2 ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) ~ 2 t2 P~ P~ + Wσt (~vj) W (∇E ) e e Φ (V.67) 2 σt2 P~ +~k j,σt1 σ σ t2 t2 ∗ σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) ~ 2 t2 P~ P~ − Wσt (~vj) W (∇E ) e e Φ . (V.68) 2 σt2 P~ j,σt1 We next prove that

ρ k(V.66)k , k(V.67) − (V.68)k ≤ O((σt1 ) t1 ln t2 ) (V.69) for some ρ > 0. To this end, we use: i) The H¨olderregularity of Φσ and ∇~ Eσ described under condition ( 2) in Theorem P~ P~ I III.1. ii) The regularity of the phase function

σ ~ t2 γσ (~vj, ∇E , t1) (V.70) t2 P~ with respect to P~ ∈ S expressed in the following estimate, which is similar to (A.3) in Lemma A.1: For ~q ∈ R3 with |~q| < s(1−θ),

σ σ 1 00 ~ t2 ~ t2 ~ − (1−δ ) (1−θ) γσ (~vj, ∇E , t1) − γσ (~vj, ∇E , t1) ≤ O( |k| 4 t ) , (V.71) t2 P~ t2 P~ +~k 1 where 0 < θ(< 1) can be chosen arbitrarily close to 1. iii) The estimate ~ σ 1σ,Λ(k) k b~ Ψ ~ k ≤ C (V.72) k,λ P |~k|3/2 from (I 5) in Theorem III.1 for P~ ∈ S, which implies Z 1/2 σ 3 σ 21/2 1/2 k Nf ΨP~ k = d k k b~k,λ ΨP~ k ≤ C | log σ | . (V.73)

Likewise, Z 1/2 σ 3 ~ −3/2 σ 21/2 k Nf ΦP~ k = d k b~k,λ + O(|k| ) ΨP~ BΛ\Bσ ≤ C | log σ |1/2 , (V.74) INFRAPARTICLE STATES IN QED I. 47

σ which controls the expected photon number in the states {Φ t1 }. As a side remark, P~ 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.

(t1) iv) The cell decomposition G is determined by t1 < t2. Moreover, since β(> 1) can −β be chosen arbitrarily large and independent of , σt1 = t1 can be made as small as desired. We ﬁrst prove the bound on k(V.66)k stated in (V.69). To this end, we use

σ σ ~ t2 t2 −i(|~k|t1−~k·~x) iγσ (~vj ,∇E ,t1) −iE t1 (t1) e − I e t2 P~ e P~ Φ (V.75) j,σt1 σ σ ~ t2 t2 iγσ (~vj ,∇E ,t1) −iE t1 −i(|~k|t −~k·~x) (t1) = e t2 P~ +~k e P~ +~k (e 1 − I)Φ (V.76) j,σt1 σ σ t2 t2 iγσ (~vj ,∇~ E ,t1) −iE t1 (t ) + e t2 P~ +~k e P~ +~k Φ 1 (V.77) j,σt1 σ σ t2 t2 iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) − e t2 P~ e P~ Φ . (V.78) j,σt1 σ The H¨older regularity of Φ t1 from i) yields P~

1 0 3 ( 4 −δ ) − 2 k (V.76) k ≤ O( t1 σt1 t1 ) , (V.79) where δ0 can be chosen arbitrarily small, and independently of . The derivation of a similar estimate is given in the proof of Theorem A.3 in the Appendix, starting from (A.27), where σ σ we refer for details. The H¨older continuity of E t2 and ∇~ E t2 , again from i), combined with P~ P~ ii), with θ suﬃciently close to 1, implies that, with ~k ∈ B , σt1

1 3 5 − 2 k (V.77) − (V.78) k ≤ O( t1 σt1 t1 ) , (V.80) as desired.

To prove the bound on sup k (V.67) − (V.68) k (V.81)

~k∈Bσ t1 stated in (V.69), we use that

σ σ σ σ σ W ∗ (∇~ E t2 ) − W ∗ (∇~ E t2 ) = W ∗ (∇~ E t2 )(W ∗ (∇~ E t2 − ∇~ E t2 ) − I) . (V.82) σt2 P~ +~k σt2 P~ σt2 P~ σt2 P~ +~k P~ We apply the Schwarz inequality in the form

σ σ (W ∗ (∇~ E t2 − ∇~ E t2 ) − I) Φe (V.83) σt2 P~ +~k P~ 3 Z d q 1/2 σ σ ≤ C ∇~ E t2 − ∇~ E t2 k N 1/2Φ k 3 P~ +~k P~ f e Bσ |~q| t2 48 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

σ σ t2 t2 iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) where in our case, Φ ≡ e t2 P~ e P~ Φ . We have e j,σt1

σ kN 1/2 Φ t1 k ≤ c | log σ |1/2 ≤ c0 ( log t )1/2 , (V.84) f P~ t1 1 as a consequence of iii). Due to i),

σ σ 1 −δ00 sup ∇~ E t2 − ∇~ E t2 ≤ O( σ 4 ) (V.85) P~ +~k P~ t1 ~k∈Bσ t1 where δ00 > 0 is arbitrarily small, and independent of (see (III.14)). Therefore,

ρ0 sup (V.67) − (V.68) ≤ O((ln t2)(σt1 ) ) ~k∈Bσ t1

0 for some ρ > 0 which does not depend on (recalling that t1 < t2).

We may now return to (V.53). From iv), and the fact that the number of cells is 3 N(t1) ≈ t1 , summation over all cells yields

N(t1) X ln(t2) (V.51) − (V.52) ≤ O( ρ ) (V.86) t j=1 1 for some ρ > 0, provided that β is suﬃciently large. This agrees with (V.37). PN(t1) The sum j=1 k (V.57) k can be treated in a similar way.

Analysis of Step b)

We have to show that the norm diﬀerence of the two vectors corresponding to the change (V.49) in (V.46) is bounded by the r.h.s. of (V.5). The proof is similar to the one just given for step a), and we shall not reiterate it. The argument is based on the same properties i) – iv) as used there.

Analysis of Step c)

Finally, we prove that the diﬀerence of the vectors corresponding to (V.50) satisﬁes

N(t ) 1 σ σ t1 t1 2 X σt ∗ σt σt iγσ (~vj ,∇~ E ,t1) −iE t1 (t1) 1 1 ~ 1 t1 P~ P~ Wσt (~vj, t1) W |σt (~vj)W |σt (∇E ) − I e e ψ 1 2 2 P~ j,σt1 j=1 2 ρ ≤ O (ln(t2)) /t1 , (V.87) INFRAPARTICLE STATES IN QED I. 49 where we deﬁne

σt1 ∗ W |σt (~vj) := W (~vj)Wσ (~vj) , (V.88) 2 σt1 t2 σ σ σ σ ∗ t1 ~ t1 ∗ ~ t1 ~ t1 W |σt (∇E ) := W (∇E ) Wσ (∇E ) . (V.89) 2 P~ σt2 P~ t1 P~ We separately discuss the diagonal and oﬀ-diagonal contributions to (V.87) from the sum over cells in G(t1).

• The diagonal terms in (V.87).

To bound the diagonal terms in (V.87), we use that

σt1 ∗ σt1 σt1 σt1 σt1 W |σ (~v )W |σ (∇~ E ) = W |σ (~v − ∇~ E ) , (V.90) t2 j t2 P~ t2 j P~

~ σt and that, by deﬁnition, ~vj ≡ ∇E 1 | ~ ~ ∗ . Using the Schwarz inequality, we obtain an P =Pj estimate analogous to (V.83). Then, we note that

σ −( 1 −δ00) sup |∇~ E t1 − ~v | ≤ O( t 4 ) , (V.91) P~ j 1 ~ (t1) P ∈Gj by the H¨older regularity of ∇~ Eσ , due to condition ( 2) in Theorem III.1; see (III.14). P~ I σ Moreover, we use (V.84) to bound the expected photon number in the states {Ψ t1 }. P~ This shows that the sum of diagonal terms can be bounded by

1 00 (t1) 2 −( 4 −δ ) −ρ O( N(t1) kψ k (ln t2) t ) ≤ O( t ln t2 ) (V.92) j,σt1 1 1

3 (t1) 2 −3 for some ρ > 0, using N(t1) = O(t ), and kψ k = O(t ). 1 j,σt1 1

• The oﬀ-diagonal terms in (V.87).

Next, we bound the off-diagonal terms in (V.87), corresponding to the inner product of vectors supported on cells j 6= l of the partition G (t1). Those are similar to the off-diagonal 1 terms Mcl,j(t, s) in (IV.6) that were discussed in detail previously. Correspondingly, we can apply the methods developed in Section IV.1, up to some modifications which we explain now. Our goal is to prove the asymptotic orthogonality of the off-diagonal terms in (V.87). We first of all prove the auxiliary result

σ σ σt ∗ t1 ~ t1 −iH 1 s (t1) lim k~aσt (~ηl,j)(s) W |σt (∇E )e ψj,σ k = 0 . (V.93) s→+∞ 1 2 P~ t1 To this end, we compare

∗ σt1 σt1 ~ (t ) ~ W |σt2 (∇E ~ )1 1 (P ) (V.94) P Gj 50 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

(t1) (where 1 (t1) is the characteristic function of the cell Gj ) to its discretization: Gj ¯ (t¯) (t) (t) 1. We pick t large enough such that G is a sub-partition of G ; in particular, Gj = PM (t¯) N(t¯) m(j)=1 Gm(j), where M = N(t) . σ ¯ ~ t1 ~ ∗ (t) 2. Furthermore, deﬁning ~um(j) := ∇E ~ ∗ , where Pm(j) is the center of the cell Gm(j), Pm(j) ~ (t¯) we have, for P ∈ Gm(j),

σ 1 ( 1 −δ00) |~u − ∇~ E t1 | ≤ C 4 , (V.95) m(j) P~ t¯

where C is uniform in t1. 3. We deﬁne M σt1 X ∗ σt1 (t¯) ~ Wσt2 (M) := W |σt2 (~um(j)) 1 (P ) (V.96) Gm(j) m(j)=1 and rewrite the vector

σ σ σt ∗ t1 ~ t1 −iH 1 s (t1) ~aσt (~ηl,j)(s) W |σt (∇E )e ψ (V.97) 1 2 P~ j,σt1 in (V.93) as Z X 3 −i~k·~x ∗ σt1 σt1 σt1 d k e W |σ (∇~ E ) − σ (M) × (V.98) t2 P~ W t2 Bκ\Bσ λ t1 σ ~ −iE t1 s (t ) ~ ∗ i|k|s ~ 1 × ~ηl,j(k) · ~ε b~ e e P ψ ~k,λ k,λ j,σt1 M Z X σt X ~ + W ∗| 1 (~u ) d3k ~η (~k) · ~ε ∗ a ei|k|s × σt2 m(j) l,j ~k,λ ~k,λ Bκ\Bσ m(j)=1 λ t1 σ −iE t1 s (t¯) × e P~ ψ . (V.99) m(j),σt1

We now observe that, at fixed t1, (III.17) and the bound (V.95) imply that the vector in (V.98) converges to the zero vector as t¯ → +∞, uniformly in s. Moreover, the norm of the vector in (V.99) tends to zero, as s → +∞, at fixed t¯. This proves (V.93). The main difference between (V.87) and the similar expression in (IV.22) that is dif- ferentiated in s is the operator

σt1 ∗ σt1 σt1 W |σ (~v )W |σ (∇~ E ) , (V.100) t2 j t2 P~ which is absent in (IV.22). To control it, we ﬁrst note that the Hamiltonian Z ⊕ 3 Hbσt := Hb ~ d P, (V.101) 1 P ,σt1 where P~ − P~ f + α1/2A~ 2 >σt1 σt1 f HbP~ ,σ := + H>σt (V.102) t1 2 1 INFRAPARTICLE STATES IN QED I. 51 with Z f ∗ 3 P~ := ~k b b~ d k , (V.103) >σt1 ~k,λ k,λ 3 \Bσ R t1 and Z f ∗ 3 H := |~k| b b~ d k , (V.104) >σt1 ~k,λ k,λ 3 \Bσ R t1 satisﬁes σ σ σ σ H t1 Ψ t1 = E t1 Ψ t1 , (V.105) bP~ P~ P~ P~ and σ σ σ t1 ∗ t1 ~ t1 [W |σ (~vj)W |σ (∇E ) , Hσ ] = 0 . (V.106) t2 t2 P~ b t1 Using (V.106), the vector in (V.87) corresponding to the j-th cell can be written as σ t1 iHσ s iγσ (~vj ,∇E ,s) −iHσ s σt σt (t) b t1 t1 P~ b t1 1 ~ 1 e Wσt (~vj, s) e e W |σt (~vj − ∇E ) ψ , (V.107) 1 2 P~ j,σt1 where

σt1 σt1 σt1 ∗ σt1 σt1 W |σ (~v − ∇~ E ) = W |σ (~v )W |σ (∇~ E ) . t2 j P~ t2 j t2 P~ 1 Similarly to our strategy in Section IV.1, we control Mcl,j(t, s) by integrating a suf- d 1 ﬁciently strong bound on | ds (Mcl,j(t, s))| with respect to s over [t, ∞); see (IV.16). The derivative in s of the j-th cell vector has the form σ σ t1 t1 d iHσ s iγσ (~vj ,∇~ E ,s) −iE s σt σt (t1) b t1 t1 P~ P~ 1 ~ 1 e Wσt (~vj, s) e e W |σt (~vj − ∇E ) ψj,σ ds 1 2 P~ t1 Z iHbσ s 3 = i e t1 W (~v , s) α i[H , ~x] · Σ~ (~k) cos(~k · ~x − |~k|s) d k × σt1 j bσt1 ~vj Bκ\Bσ t1 σ t1 σt −iE s iγσ (~vj ,∇E ,s) σt1 σt1 (t) × e P~ e t1 P~ W |σ (~v − ∇~ E ) ψ t2 j P~ j,σt Z iHbσ s 2 3 + i e t1 W (~v , s) α Σ~ (~k) cos(~k · ~x − |~k|s) d k · σt1 j ~vj Bκ\Bσ t1 Z ~ 3 · Σ~vj (~q) cos(~q · ~x − |~q|s)d q × (V.108) Bκ\Bσ t1 σ σt t1 iγσ (~vj ,∇E ,s) −iE s σt σt (t) t1 P~ P~ 1 ~ 1 × e e W |σt (~vj − ∇E ) ψ 2 P~ j,σt1 σt1 dγσt (~vj, ∇E ~ , s) iHbσt s 1 P + i e 1 Wσ (~vj, s) × t1 ds σ σ t1 t1 iγσ (~vj ,∇~ E ,s) −iE s σt σt (t1) t1 P~ P~ 1 ~ 1 × e e W |σt (~vj − ∇E ) ψ . 2 P~ j,σt1 Due to the similarity of this expression with (IV.22) – (IV.29), we can essentially adopt the σ analysis presented in Section IV.1. The only diﬀerence here is the operator Hb t1 instead of σ H t1 , and the additional term involving the commutator

~x ∗ σt1 σt1 [χ ( ) ,W |σ (∇~ E )] (V.109) h s t2 P~ 52 T.CHEN, J.FROHLICH,¨ AND A. PIZZO applied to the one-particle state σ σ t1 t1 iγσ (~vj ,∇~ E ,s) −iE s (t1) e t1 P~ e P~ ψ . (V.110) j,σt1 1 However, the latter tends to zero as s → +∞, at a rate of order O( sη ), for some -independent η > 0. This follows from the H¨olderregularity of ∇~ Eσ (condition ( 2) in Theorem III.1), P~ I and (III.17). Similarly, we treat the commutator (V.109) with the infrared tail (IV.37) in

~x σt1 σt1 place of χh( s ) (and with Hb replacing H ). It is then straightforward to see that we arrive at (V.5). INFRAPARTICLE STATES IN QED I. 53

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 Hout/in which are invariant under space-time transla- out/in tions, built from vectors { Ψh,κ }. out/in To this end, we first define a subspace, H˚κ , 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 H˚κ , we define spaces of scattering states of the system, where the asymptotic electron velocity is restricted to the region ~ ~ 1 {∇EP~ : |P | < 3 } . out/in We note that the choice of H˚κ 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 “hard out/in photons” from the photon cloud present in the states in H˚κ , 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; 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 coun- terpart 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, collective degrees of freedom are involved because of the emission of soft photons for arbitrarily long times. 54 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

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 appearance of collective degrees of freedom is reﬂected in the representation of the asymptotic electromagnetic algebra, which is non-Fock but only locally Fock (see Section VI.2). 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 inequivalent. Only for ∇EP~ = 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 Section III, we have constructed a scattering state with electron wave function h, and a dressing cloud exhibiting the correct behavior in the limit ~k → 0, with maximal photon frequency κ. To construct a space which is invariant under space-time translations, we may either focus on the vectors

−i~a·P~ −iHτ out e e ψh,κ , (VI.1) or on the vectors obtained from

N(t) ~ σt σt iHt X τ,~a iγσt (~vj ,∇E ~ ,t) −iE ~ t (t) s − lim e Wσ (~vj, t)e P e P ψj,σ (τ,~a) , (VI.2) t→+∞ t t j=1 where

∗ −i|~k|(t+τ) −i~k·~a Z 3 ~vj ·{~ε~ a e e − h.c.} τ,~a 1 d k k,λ ~k,λ W (~v , t) := exp α 2 , (VI.3) σt j q B \B |~k|(1 − k · ~v ) κ σt |~k| b j and Z (t) ~ −iEσt τ −i~a·P ~ ~ σt 3 ψj,σ (τ,~a) := e e P h(P )ψ d P. (VI.4) t (t) P~ Gj INFRAPARTICLE STATES IN QED I. 55

Using Theorem III.2, one straightforwardly ﬁnds that

−i~a·P~ −iHτ out e e ψh,κ (VI.5) N(t) σt σt −i~a·P~ −iHτ iHt X iγσ (~vj ,∇~ E ,t) −iE t (t) t P~ P~ = s − lim e e e Wσt (~vj, t)e e ψj,σ (VI.6) t→+∞ t j=1 N(t+τ) ~ σt+τ σt+τ iHt X τ,~a iγσt+τ (~vj ,∇E ~ ,t+τ) −iE ~ t (t+τ) = s − lim e Wσ (~vj, t)e P e P ψj,σ (τ,~a) . (VI.7) t→+∞ t+τ t+τ j=1 The two limits (VI.2) and (VI.7) 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 ~ 1 3 Hκ := ψh,κ (τ,~a): h(P ) ∈ C0 (S\Brα ) , τ ∈ R, ~a ∈ R . (VI.8)

out/in By construction, H˚κ is invariant under space-time translations. 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 for instance out/in κ. One can construct such states based on H˚κ according to the following procedure. We consider positive energy solutions of the form Z 3 d k ~ −i|k|t+i~k·~y Ft(~y) := Fb(k) e (VI.9) 2 (2π)3p|k| of the free wave equation ∂2F (~y) ∇~ · ∇~ F (~y) − t = 0 , (VI.10) ~y ~y t ∂t2 ~ ∞ 3 which exhibit fast decay in |~y| for arbitrary ﬁxed t, and where Fb(k) ∈ C0 (R \{0}). We then construct vector-valued test functions Z 3 ~ X d k ∗ λ ~ −i|k|t+i~k·~y Ft(~y) := ~ε F (k) e (VI.11) 3p ~k,λ b λ=± 2 (2π) |k| satisfying the wave equation (VI.10), with X Fb~ (~k) := ~ε ∗ F λ(~k) ∈ C∞( 3\{0} ; 3) (VI.12) ~k,λ b 0 R C λ We set A~(t, ~y) := eiHtA~(~y)e−iHt ; (VI.13) here A~(~y) is the expression in (II.12) with Λ = ∞. An asymptotic vector potential is constructed starting from LSZ (t → ±∞) limits of interpolating ﬁeld operators Z ∂F~ (~y) ∂A~(t, ~y) A~[F~ , t] := i A~(t, ~y) · t − · F~ (~y)d3y, (VI.14) t ∂t ∂t t 56 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

~ ~ with Ft as in (VI.11) for the positive-energy component, and with −Ft for the negative energy component. We deﬁne

out/in ψ ~ := s − lim ψh,F~ (t) , (VI.15) h,F t→+/−∞ where i A~[F~t,t]−A~[F~t,t] ψh,F~ (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 (~a,τ) in our notation). The existence of the limit in (VI.15) is a straightforward consequence of standard decay estimates for oscillating integrals under the assumption in (VI.12), combined with the propagation estimate (III.43). Finally, we can deﬁne the scattering subspaces as

n _ out/in o Hout/in := ψ : h(P~ ) ∈ C1(S\B ) , Fb~ ∈ C∞( 3 \ 0 ; 3) . (VI.17) h,F~ 0 rα 0 R C

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 Sections IV and V; for further details we refer to [24].

∞ 3 iHt ~x −iHt Theorem VI.1. The functions f ∈ C0 (R ), of the variable e t e , have strong limits for t → ∞ in Hout/in, namely: ~x s − lim eiHt f( ) e−iHtψout/in =: ψout/in (VI.18) h,F~ h f ,F~ t→+/−∞ t ∇~ E ~ σ where f~ := limσ→0 f(∇E ~ ). ∇EP~ P

The proof is obtained from an adaptation of the proof of Theorem A.3 in the Appendix. For the radiation ﬁeld, we have the following result.

Theorem VI.2. The LSZ Weyl operators n i A~[G~ ,t]−A~[G~ ,t] λ 2 3 −1 3 o e t t : Gb (~k) ∈ L (R , (1 + |~k| )d k) , λ = ± (VI.19) have strong limits in Hout/in:

~ ~ ~ ~ Wout/in(G~ ) := s − lim ei A[Gt,t]−A[Gt,t] . (VI.20) t→+/−∞ The limiting operators are unitary, and have the following properties: INFRAPARTICLE STATES IN QED I. 57

i) ~ ~ 0 out/in out/in 0 out/in 0 − ρ(G,G ) W (G~ )W (G~ ) = W (G~ + G~ )e 2 (VI.21) where X Z λ ρ(G,~ G~ 0) = 2iIm( Gbλ(~k)Gc0 (~k)d3k) . (VI.22) λ

ii) The mapping R 3 s −→ Wout/in(s G~ ) deﬁnes a strongly continuous, one parameter group of unitary operators.

iii) iHτ out/in ~ −iHτ out/in ~ e W (G)e = W (G−τ ) (VI.23) ~ where G−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 (VI.18). out/in •Aph as the norm closure of the *algebra generated by the unitary operators in (VI.20). out/in From (VI.21) and (VI.23), we conclude that Aph is the Weyl algebra associated to a free radiation field. Moreover, from straightforward approximation arguments applied to out/in out/in 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 algebras out/in Aph for the concrete model at hand that confirm structural features derived in [18] under general assumptions. out/in To study the infrared features of the representation of Aph , it suffices to analyze the expectation of the generators {Wout(G~ ) } of the algebra with respect to arbitrary states of out the form ψh,κ, out out ~ out ψh,κ , W (G) ψh,κ (VI.24) N(t) ~ σt σt X D iγσ (~vl,∇E ,t) −iE t (t) = lim e t P~ e P~ ψ , (VI.25) t→+∞ l,σt j=1, l=1

σt σt ∗ i A~[G~ t,t]−A~[G~ t,t] iγσ (~vj ,∇~ E ,t) −iE t (t) E t P~ P~ Wσt (~vl, t)e Wσt (~vj, t)e e ψj,σt . In the step passing from (VI.24) to (VI.25), we use Theorem III.2. One infers from the arguments developed in Section IV.1 that the sum of the oﬀ-diagonal terms, l 6= j, vanishes 58 T.CHEN, J.FROHLICH,¨ AND A. PIZZO in the limit. Therefore, N(t) D ~ σt σt X iγσt (~vj ,∇E ~ ,t) −iE ~ t (t) (VI.25) = lim e P e P ψj,σ , (VI.26) t→+∞ t j=1

~ ~ σt σt i A~[G~ t,t]−A~[G~ t,t] %~v (G) iγσ (~vj ,∇E ,t) −iE t (t) E j t P~ P~ e e e e ψj,σt , where Z ~u · ~ε ∗ 1 X ~k,λ ~ 2 λ ~ 3 %~u(G) := 2iRe α Gb (k) 3 d k . (VI.27) ~ 2 λ Bκ |k| (1 − ~u · bk) After solving an ODE analogous to (IV.9), we ﬁnd that the diagonal terms yield

Z C ~ ~ G %∇~ E (G) out out ~ out − 2 ~ ~ 2 3 ψh,κ , W (G)ψh,κ = e e P |h(P )| d P, (VI.28) where Z b~ ~ 2 3 CG~ := |G(k)| d k , (VI.29)

~ σt ~ σt ~ Here, we also use that ~vj ≡ ∇E ~ ∗ , combined with the convergence ∇E ~ → ∇EP~ (as t → ∞ Pj P and P~ ∈ S). out/in Now, we can reproduce the following results in [18]: The representation Π(Aph ) is out/in out/in given by a direct integral on the spectrum of the operator ~vas in H , deﬁned by

out/in iHt ~x −iHt f(~vas ) := lim e f( )e (VI.30) t→ +/−∞ t ∞ 3 for any f ∈ C0 (R ), of mutually inequivalent, irreducible representations. These representa- out/in out/in tions are coherent non-Fock for values ~vas 6= 0. The coherent factors, labeled by ~vas , are out/in out/in ∗ ~vas · ~ε 1 ~vas · ~ε~k,λ 1 ~k,λ α 2 and α 2 , (VI.31) ~ 3 out/in ~ 3 out/in |k| 2 (1 − ~vas · bk) |k| 2 (1 − ~vas · bk) for the annihilation and the creation part, aout/in and aout/in ∗, respectively. ~k,λ ~k,λ out/in The representation Π(Aph ) is locally Fock in momentum space. This property is equivalent to the following one: ~ κ ∞ 3 3 For any κ > 0, and Gc ∈ C0 (R \Bκ ; C ), the operator ~ ~ κ A[−Gt , t] (VI.32)

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

~ ~ κ out lim A[−Gt , t]ψh,κ = 0 . (VI.33) t→+∞ To prove this, we ﬁrst consider Theorem III.2, then

~ ~ κ out ~ ~ κ lim A[−Gt , t]ψh,κ = lim A[−Gt , t]ψh,κ(t) . (VI.34) t→+∞ t→+∞ INFRAPARTICLE STATES IN QED I. 59

Next, we rewrite the vector

N(t) σt σt iHt X iγσ (~vj ,∇~ E ,t) −iE t (t) ~ ~ κ ~ ~ κ t P~ P~ A[−Gt , t]ψh,κ(t) = e A[−Gt , 0] Wσt (~vj, t)e e ψj,σt (VI.35) j=1 as

+∞ N(t) Z σt σt d iHs X iγσ (~vj ,∇~ E ,s) −iE s (t) − { e A~[−G~ κ, s] W (~v , s)e t P~ e P~ ψ } ds (VI.36) ds s σt j j,σt t j=1 N(t) σt σt iHs X iγσ (~vj ,∇~ E ,s) −iE s (t) ~ ~ κ t P~ P~ + lim e Wσt (~vj, s)A[−Gs , 0]e e ψj,σ . (VI.37) s→+∞ t j=1

The integral in (VI.36), and the limit in (VI.37) exist. To see this, it is enough to follow the procedure in Section IV.1, taking into account that the operator

1 ~ ~ κ σt A[−Gs , 0] [H , ~x] (VI.38) (Hσt + i) is bounded, uniformly in t and s. The limit (VI.37) vanishes at ﬁxed t because of condition (I 4) in Theorem III.1. Therefore we ﬁnally conclude that the limit (VI.34) vanishes.

Li´enard-Wiechert ﬁelds generated by the charge

Now we brieﬂy explain how to obtain the result stated in (III.76). The assertion is ob- vious for the longitudinal degrees of freedom; see the deﬁnition of Fµν in (III.77). For the transverse degrees of freedom, we argue as follows. Similarly to the treatment of (VI.24), we arrive at a sum over the diagonal terms,

D Z out/in iHt 3 tr ˜ ~ −iHt out/inE lim ψ , e d y Fµν(0, ~y) δ(~y − ~x − d) e ψ t→±∞ h,F~ h,F~ N(t) D Z E X σt 3 tr ˜ ~ σt = lim ψj,σ , d y Fµν(0, ~y) δ(~y − ~x − d) ψj,σ , t→±∞ t t j=1 by exploiting the usual decay properties in time of solutions gt,~y of the free wave equation whose Fourier transform behaves like |~k|−1 for ~k → 0. Then, one uses Proposition 5.1 in [9] which identiﬁes the infrared coherent factor by showing that

~ D σ σ E 1 1σ,Λ(k) 1 σ 1/2 −1 Ψ , b~ Ψ + α 2 ~ε~ · ∇~ E ≤ α C |~k| (VI.39) P~ k,λ P~ 1 σ k,λ P~ |~k| 2 |~k| − ~k · ∇~ E P~ 60 T.CHEN, J.FROHLICH,¨ AND A. PIZZO for ~k → 0. This straightforwardly implies that N(t) n X D Z E ~ 2 σt 3 tr ˜ ~ σt | d | lim ψj,σ , d y Fµν(0, ~y) δ(~y − ~x − d) ψj,σ − (VI.40) t→±∞ t t j=1 N(t) Z D E ∇~ Eσt o X 2 σt σt P~ tr 3 −1/2 − |h(P~ )| ψ , ψ Fµν (0, d~) d P ≤ O(|d~| ) , (t) P~ P~ j=1 Gj which vanishes in the limit |d~| → ∞, as asserted in (III.76). INFRAPARTICLE STATES IN QED I. 61

Appendix A

In this part of the Appendix, we present detailed proofs of auxiliary results used in Section III.

Lemma A.1. The following estimates hold for P~ ∈ S:

(i) For t2 > t1 1,

σ 1 σ 1 ~ t2 − ~ t2 − |γσ (~vj, ∇E , (σt ) θ ) − γσ (~vl(j), ∇E , (σt ) θ )| ≤ O(|~vj − ~vl(j)|) , (A.1) t2 P~ 2 t2 P~ 2 σ σ where ~v ≡ ∇~ E t1 and ~v ≡ ∇~ E t2 . j P~ j P~

(ii) For t2 > t1 1,

σ σ 1 ~ t2 ~ t1 (1−δ) 1−θ | γσ (~vj, ∇E , t1) − γσ (~vj, ∇E , t1) | ≤ O [(σt ) 2 t + t1 σt ] . (A.2) t2 P~ t1 P~ 1 1 1

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

σ σ − θ (1−δ00) (1−θ) | γ (~v , ∇~ E t , s) − γ (~v , ∇~ E t , s) | ≤ O(s 4 s ) , (A.3) σt j P~ σt j ~ ~q P + s whenever γ (~v , ∇~ Eσt , s) is deﬁned diﬀerent from zero. σt j P~

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 III.1.

Lemma A.2. For s ≥ t 1, the estimates Z | ln σ | sup Σl (~k) cos(~k · ~x − |~k|s)d3k ≤ O( t ) , (A.4) ~vj 3 s ~x∈R Bκ\Bσt Z ~x tθ sup Σl (~k) cos(~k · ~x − |~k|s)d3k χ ( ) ≤ O( ) , (A.5) ~vj h 2 3 s s ~x∈R Bκ\B S σt hold, where l l0 l ~ X k k l0 1 Σ~v (k) := 2 (δl,l0 − )vj , (A.6) j ~ 2 ~ 2 l0 |k| |k| (1 − bk · ~vj) −β S −θ and where σt = t , στ := τ , with β > 1, 0 < θ < 1. Moreover, χh(~y) = 0 for |~y| < νmin and |~y| > νmax with 0 < νmin < νmax < 1 (see (III.15)). 62 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

Proof. To prove the estimate (A.4), we consider the variable ~x ﬁrst in the set

3 {~x ∈ R : |~x| < (1 − ρ)s, 0 < ρ < 1} .

3 {~x ∈ R : |~x| > (1 − ρ)s, 0 < ρ < 1} , we integrate by parts with respect to cos θ~k, and observe that the two functions

d(Σbl (bk)) Σl (k) and ~vj (A.10) b~vj b d cos θ~k

1 2 belong to L (S ; dΩ~k). This yields Z Σl (~k) cos(~k · ~x − |~k|s)d3k (A.11) ~vj Bκ\Bσt Z κ Z ~ ~ l sin(|k| |~x| + |k|s) ~ = Σb (bk)|θ =π d|k|dφ (A.12) ~vj ~k ~ σt |k| |~x| Z κ Z ~ ~ l sin(|k| |~x| − |k|s) ~ + Σb (bk)|θ =0 d|k|dφ (A.13) ~vj ~k ~ σt |k| |~x| Z d(Σl (k)) ~ ~ b~vj b sin(k · ~x − |k|s) − d|k|dΩ~k . (A.14) d cos θ~ ~ Bκ\Bσt k |k| |~x|

ln σt The absolute values of (A.12), (A.13), and (A.14), are all bounded above by O( s−ρs ), as one easily veriﬁes. This establishes (A.4), uniformly in ~x ∈ R3. To prove (A.5), we consider ~x in a set of the form

3 0 0 {~x ∈ R : (1 − ρ )s > |~x| > (1 − ρ)s, 0 < ρ < ρ < 1} . (A.15) INFRAPARTICLE STATES IN QED I. 63

We apply integration by parts with respect to |~k| in (A.12), (A.13), and (A.14) in the case S σt . As an example, we get for (A.12) Z l cos(κ (|~x| + s)) (A.12) = − Σb (bk)|θ =π dφ (A.16) ~vj ~k κ (|~x| + s) |~x| Z −θ l cos(t (|~x| + s)) + Σb (bk)|θ =π dφ (A.17) ~vj ~k t−θ (|~x| + s) |~x| Z κ Z cos(|~k| |~x| + |~k|s) − Σl (k)| d|k|dφ . (A.18) b~vj b θ~k=π S ~ 2 σt |k| |~x|(|~x| + s) Since ~x is assumed to be an element of (A.15), it follows that the bound (A.5) holds for (A.12). In the same manner, one obtains a similar bound for (A.13) and (A.14).

Theorem A.3. For θ < 1 suﬃciently close to 1, and s ≥ t, the propagation estimate

σt σt ~x iγσ (~vj ,∇~ E ,s) −iE s (t) χ ( )e t P~ e P~ ψ (A.19) h s j,σt σt σt σ iγσ (~vj ,∇~ E ,s) −iE s (t) − χ (∇~ E t )e t P~ e P~ ψ h P~ j,σt 1 1 ≤ c ν 3 | ln(σt)| (A.20) s t 2 holds, where ν > 0 is independent of .

Proof. Since the detailed proof of a closely related result is given in Theorem A2 of [24], we only sketch the argument.

Expressing χh (which we assume to be real) in terms of its Fourier transform χbh, we bound (A.19) by

Z σt ~x σt σt 3 −i~q·∇~ E −i~q· iγσ (~vj ,∇~ E ,s) −iE s (t) k d q χ (~q)(e P~ − e s ) e t P~ e P~ ψ k (A.21) bh j,σt

σt σt Z σt i(E −E )s σt 3 −i~q·∇~ E P~ ~ ~q iγσ (~vj ,∇~ E ,s) (t) ≤ k d q χ (~q)(e P~ − e P + s ) e t P~ ψ k (A.22) bh j,σt

σ σ Z i(E t −E t )s ~x σt 3 ~ ~ q −i~q· iγσ (~vj ,∇~ E ,s) (t) +k d q χ (~q) e P P + s (e s − 1) e t P~ ψ k . (A.23) bh j,σt We split the integration domains of (A.22) and (A.23) into the two regions

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

In both (A.22) and (A.23), the contribution to the integral from I+ is controlled by the decay properties of χb(~q), and one easily derives the bound in (A.20). For the contributions to (A.22) from the integral over I−, the existence of the gradient of the energy, the H¨older 64 T.CHEN, J.FROHLICH,¨ AND A. PIZZO ~ property in P of the gradient, and the decay properties of χb(~q) are enough to produce the bound in (A.20). To control (A.23), we note that the two vectors

σ σ −i~q· ~x σ Ψ t and Ψ t := e s Ψ t (A.25) ~ ~q b ~ ~q P~ P − s P − s

σt σt belong to the same ﬁber space H ~q , and that, as vectors in Fock space, Ψ and Ψ P~ − b ~ ~q P~ s P − s coincide, i.e., ~x −i~q· s σt σt I ~ ~q (e Ψ ~ ) ≡ IP~ (Ψ ~ ) . (A.26) P − s P P We split and estimate (A.23) by

σt σt Z Z i(E −E )s σt ~ ~q P~ iγσ (~vj ,∇~ E ,s) σ 3 3 P − s t P~ t (A.23) = χh(~q) e hP~ e Ψb ~ ~q d P d q (A.27) b (t) P − s I− Γj σ σ Z Z t t σ i(E ~ −E ~q )s ~ t P P~ + iγσt (~vj ,∇E ,s) σt 3 3 − χh(~q) e s h ~ e P~ Ψ d P d q (A.28) b (t) P P~ I− Γj σt σt Z Z i(E −E )s σt ~ ~q P~ iγσ (~vj ,∇~ E ,s) σ 3 3 P − s t P~ t ≤ χh(~q) e hP~ e Ψb ~ ~q d P d q (A.29) b (t) P − s I− Γj σt σt Z Z i(E −E )s σt ~ ~q P~ iγσ (~vj ,∇~ E ,s) σ 3 3 P − s t P~ t − χh(~q) e hP~ e Ψ ~ ~q d P d q b (t) P − s I− Γj σt σt Z Z i(E −E )s σt ~ ~q P~ iγσ (~vj ,∇~ E ,s) σ 3 3 P − s t P~ t + χh(~q) e hP~ e Ψ ~ ~q d P d q (A.30) b (t) P − s I− Γj σt σt σt Z Z i(E −E )s iγσ (~vj ,∇~ E ,s) ~ ~q P~ t ~ ~q σ 3 3 P − s P − s t − χh(~q) e hP~ − ~q e Ψ ~ ~q d P d q b (t) s P − s I− Γj σt σt σt Z Z i(E −E )s iγσ (~vj ,∇~ E ,s) ~ ~q P~ t ~ ~q σ 3 3 P − s P − s t + χh(~q) e hP~ − ~q e Ψ ~ ~q d P d q (A.31) b (t) s P − s I− Γj σ σ Z Z t t σ i(E ~ −E ~q )s ~ t P P~ + iγσt (~vj ,∇E ,s) σt 3 3 − χh(~q) e s h ~ e P~ Ψ d P d q . b (t) P P~ I− Γj The terms (A.29), (A.30), and (A.31) can be bounded by

Z Z 1 2 σt σt 2 3 2 3 (A.29) ≤ |χ (~q)| |h | kI (Ψ ) − I ~q (Ψ )k d P d q , (A.32) h P~ P~ P~ P~ − ~ ~q F b (t) s P − s I− Γj and

Z Z σt 1 iγσ (~vj ,∇~ E ,s) 2 σ 2 3 3 t P~ t 2 (A.30) ≤ |χh(~q)| |∆ ~q (hP~ e )| kIP~ − ~q (Ψ ~ ~q )kF d P d q , (A.33) b (t) s s P − s I− Γj and Z Z 1 2 σt 2 3 2 3 (A.31) ≤ |χh(~q)| |h ~ | kI ~ (Ψ ~ )kF d P d q , (A.34) b j P P P I− O ~q s where ~ σt σt σt iγσ (~vj ,∇E ,s) iγσ (~vj ,∇~ E ,s) iγσ (~vj ,∇~ E ,s) t ~ ~q t P~ t P~ P − s ∆ ~q (hP~ e ) := hP~ e − h ~ ~q e , (A.35) s P − s INFRAPARTICLE STATES IN QED I. 65

~q ~q ~q j (t) (t) , s (t) (t) , s (t) , s ~q (t) and O ~q := (Γj ∪ Γj ) \ (Γj ∩ Γj ), where Γj is the translate by s of the cell Γj . s 1 Using (A.3), the C −regularity of hP~ , and the deﬁnition of I−, one readily shows that the terms (A.33), (A.34) satisfy the bound (A.20), as desired. To estimate (A.32), we use the inequality

σt σt kI Ψ − I ~q Ψ k (A.36) P~ P~ P~ − ~ ~q F s P − s σt σt σt σt ≤ kI W (∇~ E )Ψ − I ~q W (∇~ E )Ψ k (A.37) P~ σt P~ P~ P~ − σt ~ ~q ~ ~q F s P − s P − s ∗ σt ∗ σt σt σt +kI ~q (W (∇~ E ) − W (∇~ E )) W (∇~ E )Ψ k , (A.38) P~ − σt P~ σt ~ ~q σt ~ ~q ~ ~q F s P − s P − s P − s where it is clear that W (∇~ Eσt )Ψσt = Φσt . (A.39) σt P~ P~ P~ Moreover, we use properties (I 2), (I 5) in Theorem III.1, where we recall that (I 2) H¨olderregularity in P~ ∈ S uniformly in σ ≥ 0 holds in the sense of

σ σ 1 −δ0 0 ~ 4 kΦP~ − ΦP~ +∆P~ kF ≤ Cδ |∆P | (A.40) and σ σ 1 −δ00 ~ ~ 00 ~ 4 |∇EP~ − ∇EP~ +∆P~ | ≤ Cδ |∆P | , (A.41) 00 0 1 ~ ~ ~ 0 00 for any 0 < δ < δ < 4 , with P, P + ∆P ∈ S, and where Cδ and Cδ are finite constants depending on δ0 and δ00, respectively. We can bound (A.37) by use of (A.40). In order to bound (A.38), we recall the definition of the Weyl operator Z ∇~ Eσ σ 1 X 3 P~ ∗ W (∇~ E ) := exp α 2 d k · (~ε b − h.c.) , (A.42) σ P~ 3 ~k,λ ~k,λ ~ 2 λ BΛ\Bσ |k| δP~ ,σ(bk) and we note that 1 Z 2 (A.38) ≤ c ∇~ Eσt − ∇~ Eσt 3 3 + d3k k b Φσ k2 (A.43) P~ ~ ~q Rt Rt ~k,λ P~ − ~q F P − s s BΛ\Bσt from a simple application of the Schwarz inequality, where Z 3 1 3 d k 2 1 := = O(| ln σt| 2 ) . (A.44) Rt ~ 3 BΛ\Bσt |k| Moreover, we have Z 3 σ 2 d k k b~k,λΦ ~ ~q kF ≤ c | ln σt| , (A.45) P − s BΛ\Bσt which is derived similarly as (V.74). From Höldercontinuity of ∇~ Eσ in P~ , (A.41), we obtain a contribution to the upper P~ bound on (A.38) which exhibits a power law decay in s. 66 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

We conclude that (A.32) is bounded by (A.20), as claimed.

Remark: By a similar procedure, one ﬁnds that for t2 ≥ s ≥ t1,

σ σ t2 t2 ~x iγt (~vj ,∇~ E ,s) −iE s (t1) 2 P~ P~ χh( ) e e ψj,σ s t2 σ σ t2 t2 σt iγt (~vj ,∇~ E ,s) −iE s (t1) 1 ln(σt2 ) ~ 2 2 P~ P~ − χh(∇E ) e e ψj,σ ≤ c . (A.46) P~ t2 sν 3/2 t1 Analogous extensions hold for the estimates in the next theorem.

Theorem A.4. Both +∞ ~x Z σ S dγ (~v , , s) ~ σt σt iH t s h σ ~x σt j s i iγσ (~vj ,∇E ,s) −iE s σ (t) e W (~v , s) J | s (s)χ ( )− b e t P~ e P~ (E t +i) ψ ds σt j σt h P~ j,σt t s ds (A.47) and +∞ ~x Z σ dγ (~v , , s) ~ σt σt iH t s n σt j s iγσ (~vj ,∇E ,s) −iE s σ (t) e W (~v , s) b e t P~ e P~ (E t + i) ψ (A.48) σt j P~ j,σt t ds ~ σt dγσ (~vj, ∇E , s) ~ σt σt t P~ iγσ (~vj ,∇E ,s) −iE s σ (t) o − e t P~ e P~ (E t + i) ψ ds ds P~ j,σt are bounded by

1 2 − 3 | ln(σ )| t 2 , (A.49) tη t ~x σt S dγσ (~vj ,∇~ E ,s) σs dγbσt (~vj , s ,s) t P~ where η > 0 is -independent. J |σt (s), ds , and ds are deﬁned in (IV.36), (IV.37), and (IV.7) – (IV.8), respectively.

Proof. S We recall from (IV.36) that for σs > σt,

S Z 1 σs σt ~ ~ ~ ~ 3 J | (s) = α i[H , ~x] · Σ~v (k) cos(k · ~x − |k|s)d k , σt j σt B S \Bσ H + i σs t S 1 where σs := sθ is the slow cut-oﬀ, and from (IV.37)

~x iHσt s −iHσt s dγˆ (~v , , s) σ 1 d{e ~x (s)e } σ σt j s := α e−iH t s h eiH t s · ds Hσt + i ds Z · Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s)d3k . (A.50) ~vj P~ B S \Bσ σs t S For s such that σs ≤ σt the expressions (A.47) and (A.48) are identically zero. By unitarity iHσt s of e and Wσt (~vj, s), we can replace the part in the integrand of (A.47) proportional to INFRAPARTICLE STATES IN QED I. 67

S σs J |σt (s) by

σ iH t s σt 1 ~x e Wσ (~vj, s)α i[H , ~x] χh( ) · (A.51) t Hσt + i s

Z σt σt 3 σ iγσ (~vj ,∇~ E ,s) −iE s σ (t) · d k Σ~ (~k) cos(~k · ∇~ E t s − |~k|s) e t P~ e P~ (E t + i)ψ , ~vj P~ P~ j,σt B S \Bσ σs t up to a term which yields an integral bounded in norm by

1 2 − 3 | ln(σ )| t 2 . (A.52) tη t To justify this step, we exploit the fact that the operator 1 i[Hσt , ~x] (A.53) Hσt + i is bounded. Moreover, we are applying the propagation estimate n Z Σ~ (~k) cos(~k · ~x − |~k|s)d3k (A.54) ~vj B S \Bσ σs t Z σt σt σ 3 o iγσ (~vj ,∇~ E ,s) −iE s σ (t) − Σ~ (~k) cos(~k · ∇~ E t s − |~k|s)d k e t P~ e P~ (E t + i) ψ ~vj P~ P~ j,σt B S \Bσ σs t 1 1 ≤ c 1+ν 3 | ln(σt)| , s t 2 for some ν > 0, which is similar to (A.19). To obtain the upper bound, we exploit the fact S S −θ σs that due 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 (IV.35), in order to use the result in Lemma A.2. Next, we approximate (A.51) by

iHσt s −iHσt s 1 d~x(s) ~x(s) e Wσ (~vj, s)α e χh( ) · (A.55) t Hσt + i ds s

Z σt 3 σ σ iγσ (~vj ,∇~ E ,s) (t) · d k Σ~ (~k) cos(~k · ∇~ E t s − |~k|s)(E t + i)e t P~ ψ ~vj P~ P~ j,σt B S \Bσ σs t where ~x(s) := eiHσt s~xe−iHσt s. To pass from (A.51) to (A.55), we have used d~x(s) 1 1 d~x(s) 1 d[~x(s) ,Hσt ] 1 = − , (A.56) ds Hσt + i Hσt + i ds Hσt + i ds Hσt + i and we have noticed that the term containing 1 d[~x(s) ,Hσt ] 1 (A.57) Hσt + i ds Hσt + i can be neglected because an integration by parts shows that the corresponding integral is 1 3 2 − 2 bounded in norm by tν | ln(σt)| t . This uses Z | ln σt)| sup d3k Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s) ≤ O (A.58) ~vj P~ P~ ∈S B S \Bσ s σs t 68 T.CHEN, J.FROHLICH,¨ AND A. PIZZO and d Z 1 sup d3k Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s) ≤ O , (A.59) ~vj P~ 1+θ P~ ∈S ds B S \Bσ s σs t which can be derived as in Lemma A.2. To bound the integral corresponding to (A.55), we note that up to a term whose integral d~x(s) ~x(s) is bounded in norm by (A.49), one can replace ds χh( s ) by

d σ σ eiH t s~x (s)e−iH t s , (A.60) ds h ~x where ~xh(s) := ~xχh( s ), with χh(~y) deﬁned as in Section IV.1. This is possible because

d σ σ eiH t s~x (s)e−iH t s (A.61) ds h σ ~x ~x σ = − eiH t s~x [ · ∇~ χ ( )]e−iH t s (A.62) s2 h s σ ~x σ + eiH t si[Hσt , ~x]χ ( )e−iH t s (A.63) h s σt σ ~x ~x i[H , ~x] σ + eiH t s ∇~ χ ( ) · e−iH t s (A.64) s h s 2 σt σ ~x i[H , ~x] ~x σ + eiH t s · ∇~ χ ( )e−iH t s , (A.65) s 2 h s d~x(s) ~x(s) where (A.63) corresponds to ds χh( s ). Moreover, we use the fact that the vector operator i j 1 σt x x j ~x Hσt +i i[H , ~x] is bounded, and apply the propagation estimate (A.19) to s2 ∇ χh( s ) and i to x ∇jχ ( ~x ) with appropriate modiﬁcations (see (A.72) and recall that ∇~ χ (∇~ Eσt ) = 0 for s h s h P~ P~ ∈ supp h). We observe that iHσt s −iHσt s iHσt s −iHσt s 1 d( e ~xh(s)e ) e Wσ (~vj, s) α e · (A.66) t Hσt + i ds

Z σt 3 σ σ iγσ (~vj ,∇~ E ,s) (t) · d k Σ~ (~k) cos(~k · ∇~ E t s − |~k|s)(E t + i) e t P~ ψ ~vj P~ P~ j,σt B S \Bσ σs t corresponds to

~x σ dγˆ (~v , , s) ~ σt iH t s h σt j s i iγσ (~vj ,∇E ,s) (t) e W (~v , s) e t P~ ψ . (A.67) σt j ds j,σt This immediately implies (A.47). To prove (A.48), we need to control the integral Z s¯ iHσt s −iHσt s iHσt s −iHσt s α d( e ~xh(s)e ) e Wσ (~vj, s)e · (A.68) t σt t H + i ds Z · d3k Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s)(Eσt + i)ψ(t) ds ~vj P~ P~ j,σt B S \Bσ σs t INFRAPARTICLE STATES IN QED I. 69 fors ¯ → +∞. An integration by parts with respect to s yields Z iHσt s α 3 ~ ~ e Wσ (~vj, s) ~xh(s) · d k Σ~v (k) × (A.69) t σt j H + i B S \Bσ σs t σ σ σ −iE t s σ (t) s¯ × cos(~k · ∇~ E t s − |~k|s) E t e P~ (E t + i)ψ P~ P~ P~ j,σt t Z s¯ n d iHσt s −iHσt s o iHσt s α − ( e Wσ (~vj, s)e ) e × (A.70) t σt t ds H + i Z ×~x (s) · d3k Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s) h ~vj P~ B S \Bσ σs t σ −iE t s σ (t) × e P~ (E t + i)ψ ds P~ j,σt Z s¯ iHσt s α − e Wσ (~vj, s) × (A.71) t σt t H + i n d Z o ×~x (s) · d3k Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s) h ~vj P~ ds B S \Bσ σs t σ −iE t s σ (t) × e P~ (E t + i)ψ ds. P~ j,σt

Here, we notice that Z ~x −i~q· ~x 3 ~x (s) = ~xχ ( ) = −is ∇~ χ (~q)e s d q . (A.72) h h s ch Furthermore, the operator Z ~ −i~q· ~x 3 −i ∇χch(~q)e s d q (A.73) tends to Z −i~q·∇~ Eσt ~ ~ 3 −i ∇χch(~q)e P d q (A.74) for s → ∞, if it is applied to the vectors

σt σt iγσ (~vj ,∇~ E ,s) −iE s (t) t P~ P~ e e ψj,σt , (A.75) or

Z σt σt 3 σ iγσ (~vj ,∇~ E ,s) −iE s (t) d k Σ~ (~k) cos(~k · ∇~ E t s − |~k|s)e t P~ e P~ ψ , (A.76) ~vj P~ j,σt B S \Bσ σs t or

Z σt σt n d 3 σ o iγσ (~vj ,∇~ E ,s) −iE s (t) d k Σ(~ ~k,~v ) cos(~k · ∇~ E t s − |~k|s) e t P~ e P~ ψ . (A.77) j P~ j,σt ds B S \Bσ σs t The rate of convergence of the corresponding expression in (A.48) is bounded by (A.49). 70 T.CHEN, J.FROHLICH,¨ AND A. PIZZO

Therefore, we can replace expressions (A.69),(A.70), and (A.71) by

σ σ eiH t sW (~v , s)e−iH t sα s∇~ Eσt · (A.78) σt j P~ Z σt σt s¯ 3 σ iγσ (~vj ,∇~ E ,s) −iE s (t) ~ ~ ~ ~ t ~ t P~ P~ · d k Σ~vj (k) cos(k · ∇E ~ s − |k|s)e e ψj,σ P t t B S \Bσ σs t s¯ Z n d σ σ o − ds ( eiH t sW (~v , s)e−iH t s ) α s∇~ Eσt · (A.79) σt j P~ t ds Z σt σt σ 3 iγσ (~vj ,∇~ E ,s) −iE s (t) · Σ~ (~k) cos(~k · ∇~ E t s − |~k|s)d ke t P~ e P~ ψ ~vj P~ j,σt B S \Bσ σs t s¯ Z σ σ − eiH t sW (~v , s)e−iH t sα s∇~ Eσt · (A.80) σt j P~ t n d Z o · Σ~ (~k) cos(~k · ∇~ Eσt s − |~k|s)d3k × ~vj P~ ds B S \Bσ σs t σt σt iγσ (~vj ,∇~ E ,s) −iE s (t) t P~ P~ × e e ψj,σt ds . Recalling the deﬁnition of the phase factor, the sum of the expressions (A.78), (A.79), and (A.80) can be written compactly as s¯ Z σ ~ σt σt iH t s dγσt (~vj, ∇EP~ , s) iγσ (~vj ,∇E ,s) −iE s (t) t P~ P~ ds e Wσt (~vj, s) e e ψj,σt , (A.81) t ds after an integration by parts. This implies the asserted bound for (A.47).

Acknowledgements. The authors gratefully acknowledge the support and hospitality of the Erwin Schr¨odinger Institute (ESI) in Vienna in June 2006, where this collaboration was initiated. T.C. was supported by NSF grants DMS-0524909 and DMS-0704031.

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Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544, USA E-mail address: [email protected]

Institut fur¨ Theoretische Physik, ETH Honggerberg,¨ CH-8093 Zurich,¨ Switzerland, and IHES,´ Bures sur Yvette, France E-mail address: [email protected]

Institut fur¨ Theoretische Physik, ETH Honggerberg,¨ CH-8093 Zurich,¨ Switzerland E-mail address: [email protected]