Infraparticle Scattering States in Non-Relativistic QED: II. Mass Shell Properties
aa r X i v : . [ m a t h - ph ] D ec Infraparticle Scattering States in Non-Relativistic QED: II. MassShell Properties
Thomas Chen
Department of Mathematics, University of Texas at Austin,1 University Station C1200, Austin, TX 78712, USA ∗ J¨urg Fr¨ohlich
Institut f¨ur Theoretische Physik, ETH H¨onggerberg,CH-8093 Z¨urich, Switzerland, and IH ´ES, Bures sur Yvette, France † Alessandro Pizzo
Department of Mathematics, One Shields Avenue,University of California Davis, Davis, CA 95616, USA ‡ Abstract
We study the infrared problem in the usual model of QED with non-relativistic matter. We provespectral and regularity properties characterizing the mass shell of an electron and one-electroninfraparticle states of this model. Our results are crucial for the construction of infraparticlescattering states, which are treated in a separate paper.
PACS numbers: 31.30.jf ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] FP Infraparticle Scattering I. INTRODUCTION
We study the dynamics of an electron interacting with the quantized electromagneticfield in the framework of non-relativistic Quantum Electrodynamics (QED). In a theorydescribing a massive particle (the electron) interacting with a field of massless bosons (thephotons), massive one-particle states do, in general, not exist in the physical Hilbert space ofthe theory. This fact was first observed by Schroer [24], who also coined the term “infraparti-cle”, a notion that generalizes that of a particle. In relativistic QED, charged infraparticleswere shown to occur, using arguments from general quantum field theory; see [3, 16]. Forthe spectrum of (
H, ~P ) in Nelson’s model, a simplified variant of non-relativistic QED, with H denoting the Hamiltonian, and ~P the total, conserved momentum of the massive particleand the massless bosons, it was proven in [14, 15] that the bottom of the spectrum of thefiber Hamiltonian H ~P at a fixed total momentum ~P ∈ R is not an eigenvalue of H ~P , forany value of ~P with | ~P | m < ρ ( λ ) < λ is the coupling constant and m is the electronmass. To prove this result, one introduces an infrared cutoff σ > H ~P turning off all interactions of the non-relativistic, massive particle with the soft modes (withfrequencies < σ ) of the relativistic, massless boson field. One then aims to establish spectralproperties of the model in the limit σ → σ~P of the infrared regularized Hamiltonian H σ~P in Nelson’s model has been developedin [22] using a novel multiscale analysis technique. In [22], important regularity propertieshave been derived, which are crucial for the analysis of the asymptotic dynamics of the elec-tron. Similarly as in [15], the strategy in [22] is to apply a specific Bogoliubov transformation to the photon variables in H σ~P , in order to obtain a Hamiltonian K σ~P whose ground state Φ σ~P remains in Fock space, as σ →
0. Subsequently, one derives properties of the ground statevector of the physical Hamiltonian H ~P in the singular limit σ → σ →
0, the latter gives rise to a coherent representationof the observable algebra of the boson field unitarily inequivalent to the Fock representationand to the coherent representations associated to different values of the total momentum.The identification of the correct Bogoliubov transformation is crucial for the constructionsin [22, 23]. For Nelson’s model, this Bogoliubov transformation has been found in [14] by amethod that exploits the linearity of the interaction in the Nelson Hamiltonian with respectto the creation- and annihilation operators. Due to the more complicated structure of theinteraction Hamiltonian in non-relativistic QED, this argument cannot be applied, and thecorrect Bogoliubov transformation for non-relativistic QED has only recently been identified
FP Infraparticle Scattering recursive analytic perturbationtheory introduced in [22], we present a new construction of the correct Bogoliubov transfor-mation, and we prove the following main results: • The ground state vectors Φ σ~P of the Bogoliubov-transformed Hamiltonians K σ~P convergestrongly to a vector in Fock space, in the limit σ →
0. The convergence rate isestimated by O ( σ η ), for some explicit η > • The vectors Φ σ~P in Fock space are H¨older continuous in ~P , uniformly in σ .These properties are key ingredients for the construction of infraparticle scattering states,which we present in [10]. A key difficulty in this analysis is the fact that the infrared behaviorof the interaction in QED is, in the terminology of renormalization group theory, of marginal type (see also [8]). FP Infraparticle Scattering II. DEFINITION OF THE MODEL
The Hilbert space of pure state vectors of the system consisting of one non-relativisticelectron interacting with the quantized electromagnetic field is given by H := H el ⊗ F , (II.1)where H el = L ( R ) is the Hilbert space for a single Schr¨odinger electron (for expositoryconvenience, we neglect the spin of the electron). The Fock space used to describe the statesof the transverse modes of the quantized electromagnetic field (the photons ) in the Coulombgauge is given by F := ∞ M N =0 F ( N ) , F (0) = C Ω , (II.2)where Ω is the vacuum vector (the state of the electromagnetic field without any excitedmodes), and F ( N ) := S N N O j =1 h , N ≥ , (II.3)where the Hilbert space h of a single photon is h := L ( R × Z ) . (II.4)Here, R is momentum space, and Z accounts for the two independent transverse polar-izations (or helicities) of a photon. In (II.3), S N denotes the orthogonal projection ontothe subspace of N Nj =1 h of totally symmetric N -photon wave functions, to account for thefact that photons satisfy Bose-Einstein statistics. Thus, F ( N ) is the subspace of F of statevectors for configurations of exactly N photons.In this paper, we use units such that Planck’s constant ~ , the speed of light c , and themass of the electron are equal to unity. The dynamics of the system is generated by theHamiltonian H := (cid:0) − i ~ ∇ ~x + α / ~A ( ~x ) (cid:1) H f . (II.5)The multiplication operator ~x ∈ R accounts for the position of the electron. The electronmomentum operator is given by ~p = − i ~ ∇ ~x . α ∼ = 1 /
137 is the finestructure constant (which,in this paper, plays the rˆole of a small parameter), ~A ( ~x ) denotes the vector potential of thetransverse modes of the quantized electromagnetic field in the Coulomb gauge , ~ ∇ ~x · ~A ( ~x ) = 0 . (II.6) FP Infraparticle Scattering H f is the Hamiltonian of the quantized, free electromagnetic field, H f := X λ = ± Z d k | ~k | a ∗ ~k,λ a ~k,λ , (II.7)where a ∗ ~k,λ and a ~k,λ are the usual photon creation- and annihilation operators, satisfying thecanonical commutation relations[ a ~k,λ , a ∗ ~k ′ ,λ ′ ] = δ λλ ′ δ ( ~k − ~k ′ ) , (II.8)[ a ~k,λ , a ~k ′ ,λ ′ ] = 0 (II.9)(where a = a or a ∗ ). The vacuum vector Ω is characterized by the condition a ~k,λ Ω = 0 , (II.10)for all ~k ∈ R and λ ∈ Z ≡ {±} .The quantized electromagnetic vector potential is given by ~A ( ~x ) := X λ = ± Z B Λ d k q | ~k | (cid:8) ~ε ~k,λ e − i~k · ~x a ∗ ~k,λ + ~ε ∗ ~k,λ e i~k · ~x a ~k,λ (cid:9) , (II.11)where ~ε ~k, − , ~ε ~k, + are photon polarization vectors, i.e., two unit vectors in R ⊗ C satisfying ~ε ∗ ~k,λ · ~ε ~k,µ = δ λµ , ~k · ~ε ~k,λ = 0 , (II.12)for λ, µ = ± . The equation ~k · ~ε ~k,λ = 0 expresses the Coulomb gauge condition. Moreover, B Λ is a ball of radius Λ centered at the origin in momentum space. Λ represents an ultravioletcutoff that will be kept fixed throughout our analysis. The vector potential defined in (II.11)is thus cut off in the ultraviolet.Throughout this paper, it will be assumed that Λ ≈ α is sufficiently small. Under these assumptions, the Hamitonian H is selfadjointon D ( H ), i.e., on the domain of definition of the operator H := ( − i ~ ∇ ~x ) H f . (II.13)The perturbation H − H is small in the sense of Kato; see, e.g., [25].The operator measuring the total momentum of the system consisting of the electron andthe electromagnetic radiation field is given by ~P := ~p + ~P f , (II.14) FP Infraparticle Scattering ~p = − i ~ ∇ ~x is the momentum operator for the electron, and ~P f := X λ = ± Z d k ~k a ∗ ~k,λ a ~k,λ (II.15)is the momentum operator associated with the photon field.The operators H and ~P are essentially selfadjoint on the domain D ( H ), and since thedynamics is invariant under translations, they commute, [ H, ~P ] = ~
0. The Hilbert space H can be decomposed on the joint spectrum, R , of the component-operators of ~P . Theirspectral measure is absolutely continuous with respect to Lebesque measure. Thus, H := Z ⊕ H ~P d P , (II.16)where each fiber space H ~P is a copy of Fock space F . Remark
Throughout this paper, the symbol ~P stands both for a variable in R and fora vector operator in H , depending on the context. Similarly, a double meaning is alsoassociated with functions of the total momentum operator. We recall that vectors Ψ ∈ H are given by sequences { Ψ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) } ∞ m =0 , (II.17)of functions, Ψ ( m ) , where Ψ (0) ( ~x ) ∈ L ( R ), of the electron position ~x and of m photonmomenta ~k , . . . , ~k m and helicities λ , . . . , λ m , with the following properties:(i) Ψ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) is totally symmetric in its m arguments ( ~k j , λ j ) j =1 ,...,m .(ii) Ψ ( m ) is square-integrable, for all m .(iii) If Ψ and Φ are two vectors in H then(Ψ , Φ) (II.18)= ∞ X m =0 (cid:0) X λ j = ± Z d x m Y j =1 d k j Ψ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) Φ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) (cid:1) . We identify a square integrable function g ( ~x ) with the sequence { Ψ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) } ∞ m =0 , (II.19)where Ψ (0) ( ~x ) ≡ g ( ~x ), and Ψ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) ≡ m >
0; analogously, asquare integrable function g ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ), m ≥
1, is identified with the sequence { Ψ ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) } ∞ m =0 , (II.20) FP Infraparticle Scattering ( m ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) ≡ g ( m ) , and Ψ ( m ′ ) ( ~x ; ~k , λ ; . . . ; ~k m , λ m ) ≡ m ′ = m .From now on, a sequence describing a quantum state with a fixed number of photons isidentified with its nonzero component wave function; vice versa, a wave function correspondsto a sequence according to the previous identification. The elements of the fiber space H ~P ∗ are obtained by linear combinations of the (improper) eigenvectors of the total momentumoperator ~P with eigenvalue ~P ∗ , e.g., the plane wave e i ~P ∗ · ~x is the eigenvector describing astate with an electron and no photon. Given any ~P ∈ R , there is an isomorphism, I ~P , I ~P : H ~P −→ F b , (II.21)from the fiber space H ~P to the Fock space F b , acted upon by the annihilation- and creationoperators b ~k,λ , b ∗ ~k,λ , where b ~k,λ corresponds to e i~k · ~x a ~k,λ , and b ∗ ~k,λ to e − i~k · ~x a ∗ ~k,λ , and with vacuumΩ f := I ~P ( e i ~P · ~x ). To define I ~P more precisely, we consider a vector ψ ( f ( n ) ; ~P ) ∈ H ~P with adefinite total momentum, ~P , describing an electron and n photons. Its wave function in thevariables ( ~x ; ~k , λ ; . . . , ~k n , λ n ) is given by e i ( ~P − ~k −···− ~k n ) · ~x f ( n ) ( ~k , λ ; . . . ; ~k n , λ n ) (II.22)where f ( n ) is totally symmetric in its n arguments. The isomorphism I ~P acts by way of I ~P (cid:0) e i ( ~P − ~k −···− ~k n ) · ~x f ( n ) ( ~k , λ ; . . . ; ~k n , λ n ) (cid:1) (II.23)= 1 √ n ! X λ ,...,λ n Z d k . . . d k n f ( n ) ( ~k , λ ; . . . ; ~k n , λ n ) b ∗ ~k ,λ · · · b ∗ ~k n ,λ n Ω f . The Hamiltonian H maps each H ~P into itself, i.e., it can be written as H = Z H ~P d P , (II.24)where H ~P : H ~P −→ H ~P . (II.25)Written in terms of the operators b ~k,λ , b ∗ ~k,λ , and of the variable ~P , the fiber Hamiltonian H ~P has the form H ~P := (cid:0) ~P − ~P f + α / ~A (cid:1) H f , (II.26)where ~P f = X λ Z d k ~k b ∗ ~k,λ b ~k,λ , (II.27) H f = X λ Z d k | ~k | b ∗ ~k,λ b ~k,λ , (II.28) FP Infraparticle Scattering ~A := X λ Z B Λ d k q | ~k | (cid:8) b ∗ ~k,λ ~ε ~k,λ + ~ε ∗ ~k,λ b ~k,λ (cid:9) . (II.29)In the following, we will only construct infraparticle states of momentum ~P ∈ S , where S := { ~P ∈ R : | ~P | < } . (II.30)(Our results can be extended to a region S (inside the unit ball) of radius larger than 1 / σ > H I, ~P := α / ~A · ( ~P − ~P f ) + α ~A H ~P , which is imposed on the vector potential ~A . Its removal is the mainproblem solved in this paper. Our results are crucial ingredients for infraparticle scatteringtheory; see [10]. We will start by studying the regularized fiber Hamiltonian H σ~P := (cid:0) ~P − ~P f + α / ~A σ (cid:1) H f (II.32)acting on the fiber space H ~P , for ~P ∈ S , where ~A σ := X λ Z B Λ \B σ d k q | ~k | (cid:8) b ∗ ~k,λ ~ε ~k,λ + ~ε ∗ ~k,λ b ~k,λ (cid:9) (II.33)and where B σ is a ball of radius σ . We will consider a sequence ( σ j ) ∞ j =0 of infrared cutoffsgiven by σ j := Λ ǫ j , with 0 < ǫ < j ∈ N := N ∪ { } .In Section IV, we construct the ground state vector (Ψ σ j ~P ) of the Hamiltionan ( H σ j ~P ), andwe compare ground state vectors Ψ σ j ~P , Ψ σ j ′ ~P ′ corresponding to different fiber Hamiltonians H σ j ~P , H σ j ′ ~P ′ with ~P = ~P ′ . We compare the vectors Ψ σ j ~P , Ψ σ j ′ ~P ′ as elements of the space F b .More precisely, we use the expression k Ψ σ j ~P − Ψ σ j ~P ′ k F (II.34)as an abbreviation for k I ~P (Ψ σ j ~P ) − I ~P ′ (Ψ σ j ~P ′ ) k F . (II.35) A. Background
In a companion paper [10], we construct a vector ψ h, Λ ( t ) converging to a scatteringstate ψ out/inh, Λ , as time t tends to infinity, applying and extending mathematical techniques FP Infraparticle Scattering ψ out/inh, Λ represents an electron with a wavefunction h in the momentum variable with support contained in S = { ~P : | ~P | < } ,accompanied by a cloud of real photons described by a Bloch-Nordsieck factor, and with anupper photon frequency cutoff Λ .In [10] we also construct the scattering subspaces H out/in , starting from certain subspaces, H out/in , and applying ”hard” asymptotic photon creation operators. These spaces carryrepresentations of the algebras, A out/inph and A out/inel , of asymptotic photon- and electronobservables, respectively, and the fact that their actions commute proves, mathematically,asymptotic decoupling of the electron and photon dynamics, as time t → ±∞ . Propertiesof the representations of A out/inph in the infrared expected on the basis of the Bloch-Nordsieckparadigm are rigorously established; see [10]. FP Infraparticle Scattering III. STATEMENT OF THE MAIN RESULTS
The main results of our paper are summarized in Theorem III.1 below. They are funda-mental for the construction of scattering states in [10] and are very similar to those used inthe analysis of Nelson’s model in [23].We define the energy of a dressed one-electron state of momentum ~P by E σ~P = inf spec H σ~P , E ~P = inf spec H ~P = E σ =0 ~P . (III.1)We refer to E σ~P as the ground state energy of the fiber Hamiltonian H σ~P . If it exists thecorresponding ground state is denoted by Ψ σ~P . We always assume that ~P ∈ S := { ~P ∈ R : | ~P | < } and that α is so small that, for all ~P ∈ S , | ~ ∇ E σ~P | < ν max < ν max , uniformly in σ . The existence of ~ ∇ E σ~P will be proven in SectionIV A.Let δ σ~P ( b k ) be given by δ σ~P ( b k ) := 1 − ~k · ~ ∇ E σ~P | ~k | . (III.3)We introduce an operator W σ ( ~ ∇ E σ~P ) := exp (cid:16) α X λ Z B Λ \B σ d k ~ ∇ E σ~P | ~k | δ σ~P ( b k ) · ( ~ε ~k,λ b ∗ ~k,λ − h.c. ) (cid:17) , (III.4)on H ~P , which is unitary for σ >
0, and consider the transformed fiber Hamiltonian K σ~P := W σ ( ~ ∇ E σ~P ) H σ~P W ∗ σ ( ~ ∇ E σ~P ) . (III.5)Conjugation by W σ ( ~ ∇ E σ~P ) acts on the creation- and annhilation operators by a (Bogoliubov)translation W σ ( ~ ∇ E σ~P ) b ~k,λ W ∗ σ ( ~ ∇ E σ~P ) = b ~k,λ − α / σ, Λ ( ~k ) | ~k | δ σ~P ( b k ) ~ ∇ E σ~P · ~ε ~k,λ , (III.6)where σ, Λ ( ~k ) stands for the characteristic function of the set B Λ \ B σ . Our methods exploitregularity properties in σ and ~P of the ground state vector, Φ σ~P , and of the ground stateenergy, E σ~P , of K σ~P . These properties are formulated in the following theorem, which is themain result of this paper. Theorem III.1.
For ~P ∈ S and for α > sufficiently small, the following statements hold. FP Infraparticle Scattering ( I ) The energy E σ~P is a simple eigenvalue of the operator K σ~P on F b . Let B σ := { ~k ∈ R | | ~k | ≤ σ } , and let F σ denote the Fock space over L (( R \ B σ ) × Z ) . Likewise, wedefine F σ to be the Fock space over L ( B σ × Z ) ; hence F b = F σ ⊗ F σ . On F σ , theoperator K σ~P has a spectral gap of size ρ − σ or larger, separating E σ~P from the rest ofits spectrum, for some constant ρ − (depending on α ), with < ρ − < .The contour γ := { z ∈ C || z − E σ~P | = ρ − σ } , σ > bounds a disc which intersects the spectrum of K σ~P | F σ in only one point, { E σ~P } . Thenormalized ground state vectors of the operators K σ~P are given by Φ σ~P := π i R γ K σ~P − z dz Ω f k π i R γ K σ~P − z dz Ω f k F (III.8) and converge strongly to a non-zero vector Φ ~P ∈ F b , in the limit σ → . The rate ofconvergence is, at least, of order σ (1 − δ ) , for any < δ < . Formula (III.8) fixes thephase of Φ σ~P such that | (Φ σ~P , Ω f ) | > .The dependence of the ground state energies E σ~P of the fiber Hamiltonians K σ~P on theinfrared cutoff σ is characterized by the following estimates. | E σ~P − E σ ′ ~P | ≤ O ( σ ) , (III.9) and | ~ ∇ E σ~P − ~ ∇ E σ ′ ~P | ≤ O ( σ (1 − δ ) ) , (III.10) for any < δ < , with σ > σ ′ > .( I ) The following H¨older regularity properties in ~P ∈ S hold uniformly in σ ≥ : k Φ σ~P − Φ σ~P +∆ ~P k F ≤ C δ ′ | ∆ ~P | − δ ′ (III.11) and | ~ ∇ E σ~P − ~ ∇ E σ~P +∆ ~P | ≤ C δ ′′ | ∆ ~P | − δ ′′ , (III.12) for < δ ′′ < δ ′ < , with ~P , ~P + ∆ ~P ∈ S , where C δ ′ and C δ ′′ are finite constantsdepending on δ ′ and δ ′′ , respectively.( I ) Given a positive number < ν min < , there are numbers ν max independent of ν min aslong as < ν min < ν max < , and r α = ν min + O ( α ) > , such that, for ~P ∈ S \ B r α and for α sufficiently small, > ν max ≥ | ~ ∇ E σ~P | ≥ ν min > , (III.13) FP Infraparticle Scattering FIG. 1: The condition ( I uniformly in σ .( I ) For ~P ∈ S and for any ~k = 0 , the following inequality holds uniformly in σ , for α small enough: E σ~P − ~k > E σ~P − C α | ~k | , (III.14) where E σ~P − ~k := inf spec H σ~P − ~k and < C α < , with C α → as α → .( I ) Let Ψ σ~P ∈ F denote the ground state vector of the fiber Hamiltonian H σ~P , so that Φ σ~P = ζ W σ ( ~ ∇ E σ~P ) Ψ σ~P k Ψ σ~P k F , ζ ∈ C , | ζ | = 1 . (III.15) For ~P ∈ S , one has that k b ~k,λ Ψ σ~P k Ψ σ~P k F k F ≤ C α / σ, Λ ( ~k ) | ~k | / , (III.16) see Lemma 6.1 of [9] which can be extended to ~k ∈ R using ( I ). The proof of statement ( I
1) is given in Section IV; the proofs of statements ( I
2) and( I
3) are presented in Section V. Statement ( I
4) is proven in Section VI. We note thatcondition ( I
4) plays an important rˆole also in atomic and molecular bound state problems,see for instance [19].We note that in Section IV B below, we will, by a slight abuse of notation, use the samesymbol Φ σ~P for the ground state vector of K σ~P without normalization. FP Infraparticle Scattering A. Remark about infrared representations
The statement ( I k b ~k,λ Ψ σ~P k F ≤ C α / σ, Λ ( ~k ) | ~k | / , (III.17)follows from the identity b ~k,λ Ψ σ~P = − α σ, Λ ( ~k ) | ~k | H ~P − ~k,σ + | ~k | − E σ~P ~ε ~k,λ · ~ ∇ ~P H σ~P Ψ σ~P (III.18)which is derived by using a standard ”pull-through argument”. Combined with the uniformbounds on the renormalized mass of the electron established in [8], it is used in [9] to provethe bound (cid:10) Ψ σ~P , N f Ψ σ~P (cid:11) := Z d k (cid:10) Ψ σ~P , b ∗ ~k,λ b ~k,λ Ψ σ~P (cid:11) ≤ Cα (1 + | ~P | | ln( σ ) | ) (III.19)on the expected number of photons in the ground state Ψ σ~P . Without using the uniformbounds on the renormalized mass, one obtains the weaker upper bound (III.17). Importantimplications of this result, analyzed in [9] and used in [10], can be summarized as follows.Let A ρ denote the C ∗ -algebra of bounded operators on the Fock space F ( L (( R \ B ρ ) × Z )), where B ρ = { ~k ∈ R | | ~k | ≤ ρ } , and let A denote the C ∗ -algebra A := W ρ> A ρ k · k op ,where the closure is taken in the operator norm. We define the state ω σ~P := h Ψ σ~P , ( · ) Ψ σ~P i on A . We will show that the weak-* limit of the family of states ω σ~P , as σ →
0, existsand defines a state ω ~P on A . A somewhat weaker result of this kind (convergence of asubsequence) has been proven in [9]. An important ingredient in [9] are the uniform boundson the renormalized electron mass established in [8].The representation of A determined by ω ~P through the GNS construction can be charac-terized as follows. Let α ~P : A → A denote the Bogoliubov automorphism defined by α ~P ( A ) = lim σ → W σ ( ~ ∇ E σ~P ) A W ∗ σ ( ~ ∇ E σ~P ) (III.20)with W σ ( ~ ∇ E σ~P ) defined in (III.4), and A ∈ A . Then the GNS representation π ~P of A isequivalent to π F ock ◦ α ~P , where π F ock denotes the Fock representation. In particular, π ~P is a coherent infrared representation unitarily inequivalent to π F ock , for ~P = ~
0, and identical to π F ock if ~P = ~
0; see also [9].
FP Infraparticle Scattering IV. PROOF OF ( I ) IN THE MAIN THEOREM In this section, we prove the statements ( I
1) in Theorem III.1. This is the most involvedpart of our analysis.In the following, we write k ψ k , instead of k ψ k F , for the norm of a vector ψ ∈ F b ∼ = H ~P .We also use the notation k A k H = k A | H k for the norm of a bounded operator A acting on aHilbert space H . Typically, H will be some subspace of F b . A. Construction of the sequence { Ψ σ j ~P } of ground states We recall the definition of the fiber Hamiltonian from (II.26), H σ j ~P = (cid:0) ~P − ~P f + α / ~A σ j (cid:1) H f . (IV.1)It acts on a fixed fiber space H ~P , with ~P ∈ S , where ~A σ j = X λ = ± Z B Λ \B σj d k q | ~k | (cid:8) ~ε ~k,λ b ∗ ~k,λ + ~ε ∗ ~k,λ b ~k,λ (cid:9) (IV.2)contains an infrared cutoff at σ j := Λ ǫ j , j ∈ N , (IV.3)with 0 < ǫ < ≈ H σ j ~P has a unique ground state Ψ σ j ~P , which we construct below using an approach developed in[22].We define the Fock spaces F σ j := F b ( L (( R \ B σ j ) × Z )) and F σ j σ j +1 := F b ( L (( B σ j \ B σ j +1 ) × Z )) . It is clear that F σ j +1 = F σ j ⊗ F σ j σ j +1 , (IV.4)and that the Hamiltonians { H σ j ~P | j ∈ N } are related to one another by H σ j +1 ~P = H σ j ~P + ∆ H ~P | σ j σ j +1 , (IV.5)where ∆ H ~P | σ j σ j +1 := α ~ ∇ ~P H σ j ~P · ~A | σ j σ j +1 + α ~A | σ j σ j +1 ) (IV.6)and ~A | σ j σ j +1 := X λ = ± Z B σj \B σj +1 d k q | ~k | (cid:8) ~ε ~k,λ b ∗ ~k,λ + ~ε ∗ ~k,λ b ~k,λ (cid:9) . (IV.7) FP Infraparticle Scattering α sufficiently small and ~P ∈ S , we construct ground state vectors { Ψ σ j ~P } of the Hamil-tonians { H σ j ~P } , j ∈ N . We will prove the following results, adapting recursive argumentsdeveloped in [22].We introduce four parameters ǫ , ρ + , ρ − , µ with the properties that0 < ρ − < µ < ρ + < − C α <
23 (IV.8)0 < ǫ < ρ − ρ + (IV.9)where C α is defined in (III.14). Then, for α small enough depending on Λ, ǫ , ρ − , µ , ρ + , weprove: • The infimum of the spectrum of H σ j ~P on F σ j , which we denote by E σ j ~P , is an isolated,simple eigenvalue which is separated from the rest of the spectrum by a gap ρ − σ j orlarger. • E σ j ~P is also the ground state energy of the operators H σ j ~P and H σ j ~P − (1 − C α ) H f | σ j σ j +1 on F σ j +1 , where H f | σ j σ j +1 is defined in Eq (IV.21). Note that E σ j ~P = inf spec H σ j ~P | F σ , for any σ ≤ σ j , and that E σ j ~P is a simple eigenvalue of H σ j ~P | F σj +1 separated by a gap ≥ ρ + σ j +1 from the rest of the spectrum. • The ground state energies E σ j ~P and E σ j +1 ~P of the Hamiltonians H σ j ~P and H σ j +1 ~P , respec-tively, acting on the same space F σ j +1 satisfy0 ≤ E σ j +1 ~P ≤ E σ j ~P + c α σ j , (IV.10)where c is a constant independent of j and α but Λ-dependent.We recursively construct the ground state vector, Ψ σ j ~P (which, at this stage, is not nor-malized), of H σ j ~P on F σ j , as follows. In the initial step, we set Ψ σ ~P = Ω f .Let Ψ σ j ~P denote the ground state of the Hamiltonian H σ j ~P on F σ j with non-degenerateeigenvalue E σ j ~P and a spectral gap at least as large as ρ − σ j . We note that E σ ~P ≡ ~P is anon-degenerate eigenvalue of H σ ~P on F σ , and thatgap( H σ ~P | F σ ) ≥ σ ≥ ρ − σ , (IV.11)where gap( H ) := inf { spec( H ) \ { inf spec( H ) } } − inf spec( H ) . (IV.12) FP Infraparticle Scattering σ j ~P ⊗ Ω f ∈ F σ j +1 = F σ j ⊗ F σ j σ j +1 , (IV.13)where k Ψ σ j ~P ⊗ Ω f k = k Ψ σ j ~P k , (IV.14)is an eigenvector of H σ j ~P | F σj +1 . In (IV.13), Ω f stands for the vacuum state in F σ j σ j +1 (if notfurther specified otherwise, Ω f denotes the vacuum state in any of the photon Fock spaces).Moreover, we note that (IV.13) is the ground state of H σ j ~P restricted to F σ j +1 , becauseinf spec (cid:16) H σ j ~P (cid:12)(cid:12) F σj +1 ⊖{ C Ψ σj~P ⊗ Ω f } − E σ j ~P (cid:17) ≥ min n ρ − σ j , inf ~k ∈ R \B σj +1 { E σ j ~P + ~k + | ~k | − E σ j ~P } , σ j +1 o ≥ min n ρ − σ j , (1 − C α ) σ j +1 o ≥ ρ + σ j +1 > , (IV.15)where F σ j +1 ⊖ { C Ψ σ j ~P ⊗ Ω f } is the orthogonal complement in F σ j +1 of the one-dimensionalsubspace { C Ψ σ j ~P ⊗ Ω f } . We use property ( I H σ j ~P | F σ ), 0 ≤ σ ≤ σ j ,with the same C α , to pass from the first to the second line, and from the second to the thirdline in (IV.15); for a proof of property ( I
4) see Section VI.Consequently, the spectral gap of H σ j ~P restricted to F σ j +1 is bounded from below bygap( H σ j ~P | F σj +1 ) ≥ ρ + σ j +1 . (IV.16)We define the contour γ σ j +1 := { z j +1 ∈ C (cid:12)(cid:12) | z j +1 − E σ j ~P | = µσ j +1 } which is the boundary ofa closed disc that contains the non-degenerate ground state eigenvalue E σ j ~P of H σ j ~P , but noother elements of the spectrum of H σ j ~P | F σj +1 ; see also Figure 2 below.Then we defineΨ σ j +1 ~P := 12 πi I γ j +1 dz j +1 H σ j +1 ~P − z j +1 Ψ σ j ~P ⊗ Ω f = X n ≥ πi I γ j +1 dz j +1 H σ j ~P − z j +1 (IV.17) (cid:16) − ∆ H ~P | σ j σ j +1 H σ j ~P − z j +1 (cid:17) n Ψ σ j ~P ⊗ Ω f , which is, by construction, the ground state eigenvector of H σ j +1 ~P | F σj +1 . The associated groundstate eigenvalue E σ j +1 ~P , with H σ j +1 ~P Ψ σ j +1 ~P = E σ j +1 ~P Ψ σ j +1 ~P , is non-degenerate by Kato’s theorem. FP Infraparticle Scattering α , we show that, for z j +1 ∈ γ j +1 ,sup z j +1 ∈ γ j +1 (cid:13)(cid:13)(cid:13) (cid:16) H σ j ~P − z j +1 (cid:17) ∆ H ~P | σ j σ j +1 (cid:16) H σ j ~P − z j +1 (cid:17) (cid:13)(cid:13)(cid:13) F σj +1 ≤ C α / ǫ / [min { ( ρ + − µ ) , µ } ] / , (IV.18)where the constant on the r.h.s. depends on ~P and Λ. The largest value of α such that(IV.18) < ǫ and µ . The estimate (IV.18) is obtained from the followingbounds, which depend critically on the spectral gap (as in the model treated in [22]):i) For z j +1 ∈ γ j +1 ,sup z j +1 ∈ γ j +1 (cid:13)(cid:13)(cid:13) (cid:16) H σ j ~P − z j +1 (cid:17) ( ~ ∇ ~P H σ j ~P ) (cid:16) H σ j ~P − z j +1 (cid:17) (cid:13)(cid:13)(cid:13) F σj +1 ≤ O (cid:16) ǫ j +1 min { ( ρ + − µ ) , µ } (cid:17) (IV.19)where the implicit constant depends on ~P and Λ.ii) Writing ( ~A | σ j σ j +1 ) − and ( ~A | σ j σ j +1 ) + for the parts in ~A | σ j σ j +1 which contain annihilation- andcreation operators, respectively, we have that k ( ~A | σ j σ j +1 ) − ψ k ≤ (cid:16) Z B σj \B σj +1 d k | ~k | (cid:17) / k ( H f | σ j σ j +1 ) / ψ k≤ c ǫ j k ( H f | σ j σ j +1 ) / ψ k , (IV.20)where H f | σ j σ j +1 := X λ Z B σj \B σj +1 d k | ~k | b ∗ ~k,λ b ~k,λ , (IV.21)with ψ in the domain of ( H f | σ j σ j +1 ) / . Moreover,0 < [( ~A | σ j σ j +1 ) − , ( ~A | σ j σ j +1 ) + ] ≤ c ′ ǫ j , (IV.22)where the constants c , c ′ are proportional to Λ / and Λ, respectively.iii) For z j +1 ∈ γ j +1 ,sup z j +1 ∈ γ j +1 (cid:13)(cid:13)(cid:13) (cid:16) H σ j ~P − z j +1 (cid:17) H f | σ j σ j +1 (cid:16) H σ j ~P − z j +1 (cid:17) (cid:13)(cid:13)(cid:13) F σj +1 ≤ O ( 1 ρ + − µ ) , (IV.23)which follows from the spectral theorem for the commuting operators H f | σ j σ j +1 and H σ j ~P (one can for instance see this by adding and subtracting a suitable fraction of H f | σ j σ j +1 in the denominator). FP Infraparticle Scattering FIG. 2: The contour integral in the energy plane.
Using (IV.18), one concludes that k Ψ σ j +1 ~P − Ψ σ j ~P k ≤ C α k Ψ σ j ~P k , (IV.24)with C uniform in j , such that, for α small enough, k Ψ σ j +1 ~P k ≥ C ′ k Ψ σ j ~P k , (IV.25)for a constant C ′ > j . In particular, the vector constructed in (IV.17) isindeed non-zero.Because of (IV.10), which follows from a variational argument, we find that, for α smallenough and Λ-dependent, but independent of j ,gap( H σ j +1 ~P | F σj +1 ) ≥ µσ j +1 − c α σ j ≥ ρ − σ j +1 . (IV.26)This estimate allows us to proceed to the next scale.It easily follows from the previous results that E σ j ~P is simple and isolated, and ( H σ j ~P ) ~P ∈S is an analytic family of type A. In particular, this allows us to express ~ ∇ E σ j ~P , as a functionof ~P , by using the Feynman-Hellman formula; see (IV.27) below. B. Transformed Hamiltonians and the sequence of ground states { Φ σ j ~P } In this section, we consider the Hamiltonians obtained from { H σ j ~P } after a j − dependentBogoliubov transformation of the photon variables. In the limit j → ∞ , this transformation FP Infraparticle Scattering H ~P has a ground state.
1. Bogoliubov transformation and canonical form of the Hamiltonian
The Feynman-Hellman formula yields ~ ∇ E σ j ~P = ~P − (cid:10) ~P f − α / ~A σ j (cid:11) Ψ σj~P , (IV.27)where (cid:10) ~P f − α / ~A σ j (cid:11) Ψ σj~P := (cid:10) Ψ σ j ~P , ( ~P f − α / ~A σ j ) Ψ σ j ~P (cid:11)(cid:10) Ψ σ j ~P , Ψ σ j ~P (cid:11) . (IV.28)We define ~β σ j := ~P f − α / ~A σ j δ σ j ~P ( b k ) := 1 − b k · ~ ∇ E σ j ~P c ∗ ~k,λ := b ∗ ~k,λ + α ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) c ~k,λ := b ~k,λ + α ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) . (IV.29)We then rewrite H σ j ~P as H σ j ~P = (cid:0) ~P − ~β σ j (cid:1) H f , (IV.30)and ~P = ~ ∇ E σ j ~P + h ~β σ j i Ψ σj~P , (IV.31)thus obtaining H σ j ~P = ~P − ( ~ ∇ E σ j ~P + h ~β σ j i Ψ σj~P ) · ~β σ j + ( ~β σ j ) H f = ~P ~β σ j ) − h ~β σ j i Ψ σj~P · ~β σ j + X λ Z R \ ( B Λ \B σj ) | ~k | δ σ j ~P ( b k ) b ∗ ~k,λ b ~k,λ d k + X λ Z B Λ \B σj | ~k | δ σ j ~P ( b k ) c ∗ ~k,λ c ~k,λ d k (IV.32) − α X λ Z B Λ \B σj | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) d k . FP Infraparticle Scattering h ~β σ j i σj~P , one gets H σ j ~P = ~P − h ~β σ j i σj~P (cid:0) ~β σ j − h ~β σ j i Ψ σj~P (cid:1) X λ Z R \ ( B Λ \B σj ) | ~k | δ σ j ~P ( b k ) b ∗ ~k,λ b ~k,λ d k + X λ Z B Λ \B σj | ~k | δ σ j ~P ( b k ) c ∗ ~k,λ c ~k,λ d k (IV.33) − α X λ Z B Λ \B σj | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) d k . Next, we apply the Bogoliubov transformation b ∗ ~k,λ −→ W σ j ( ~ ∇ E σ j ~P ) b ∗ ~k,λ W ∗ σ j ( ~ ∇ E σ j ~P ) = b ∗ ~k,λ − α ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) b ~k,λ −→ W σ j ( ~ ∇ E σ j ~P ) b ~k,λ W ∗ σ j ( ~ ∇ E σ j ~P ) = b ~k,λ − α ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) (IV.34)for ~k ∈ B Λ \ B σ j , where W σ j ( ~ ∇ E σ j ~P ) := exp (cid:16) α X λ Z B Λ \B σj d k ~ ∇ E σ j ~P | ~k | δ σ j ~P ( b k ) · ( ~ε ~k,λ b ∗ ~k,λ − h.c. ) (cid:17) . (IV.35)It is evident that W σ j acts as the identity on F b ( L ( B σ j × Z )) and on F b ( L (( R \B Λ ) × Z )).Moreover, we define the vector operators ~ Π σ j ~P := W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) − h W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) i Ω f , (IV.36)noting that h ~β σ j i Ψ σj~P = ~P − ~ ∇ E σ j ~P (IV.37)= (cid:10) Φ σ j ~P , ~ Π σ j ~P Φ σ j ~P (cid:11)(cid:10) Φ σ j ~P , Φ σ j ~P (cid:11) + h W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) i Ω f , where Φ σ j ~P is the ground state of the Bogoliubov-transformed Hamiltonian K σ j ~P := W σ j ( ~ ∇ E σ j ~P ) H σ j ~P W ∗ σ j ( ~ ∇ E σ j ~P ) . (IV.38)We remark that although we have not specified the phase ζ in (III.15) yet, the expression in(IV.37) is uniquely defined, since it does not depend on ζ and on the normalization of Φ σ j ~P . FP Infraparticle Scattering W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) − h ~β σ j i Ψ σj~P = ~ Π σ j ~P − h ~ Π σ j ~P i Φ σj~P . (IV.39)As in [22], it is convenient to write K σ j ~P in the ”canonical” form K σ j ~P = (cid:0) ~ Γ σ j ~P ) X λ Z R | ~k | δ σ j ~P ( b k ) b ∗ ~k,λ b ~k,λ d k + E σ j ~P , (IV.40)where ~ Γ σ j ~P := ~ Π σ j ~P − (cid:10) ~ Π σ j ~P (cid:11) Φ σj~P , (IV.41)so that (cid:10) ~ Γ σ j ~P (cid:11) Φ σj~P = 0 , (IV.42)and E σ j ~P := ~P − ( ~P − ~ ∇ E σ j ~P ) − α X λ Z B Λ \B σj | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) d k . One arrives at (IV.40) using W σ j ( ~ ∇ E σ j ~P ) c ∗ ~k,λ W ∗ σ j ( ~ ∇ E σ j ~P ) = b ∗ ~k,λ ,W σ j ( ~ ∇ E σ j ~P ) c ~k,λ W ∗ σ j ( ~ ∇ E σ j ~P ) = b ~k,λ , (IV.44)for ~k ∈ B Λ \B σ j . The Hamiltonian K σ j ~P has a structure similar to the Bogoliubov-transformedNelson Hamiltonian in [22].Following ideas of [22], we define the intermediate Hamiltonian b K σ j +1 ~P := W σ j +1 ( ~ ∇ E σ j ~P ) H σ j +1 ~P W ∗ σ j +1 ( ~ ∇ E σ j ~P ) , (IV.45)where W σ j +1 ( ~ ∇ E σ j ~P ) := exp (cid:16) α X λ Z B Λ \B σj +1 d k ~ ∇ E σ j ~P | ~k | δ σ j ~P ( b k ) · ( ~ε ~k,λ b ∗ λ ( ~k ) − h.c. ) (cid:17) , (IV.46)and split it into different terms similarly as for K σ j ~P . We write H σ j +1 ~P = ~P − ~P · ~β σ j +1 + ( ~β σ j +1 ) H f , (IV.47) FP Infraparticle Scattering ~P by ~ ∇ E σ j ~P + h ~β σ j i Ψ σj~P , thus obtaining H σ j +1 ~P = ~P − (cid:0) ~ ∇ E σ j ~P + h ~β σ j i Ψ σj~P (cid:1) · ~β σ j +1 + ( ~β σ j +1 ) H f = ~P ~β σ j +1 ) − h ~β σ j i Ψ σj~P · ~β σ j +1 + X λ Z R \ ( B Λ \B σj +1 ) | ~k | δ σ j ~P ( b k ) b ∗ ~k,λ b ~k,λ d k + X λ Z B Λ \B σj +1 | ~k | δ σ j ~P ( b k ) c ∗ ~k,λ c ~k,λ d k (IV.48) − α X λ Z B Λ \B σj +1 | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) d k . We add and subtract h ~β σ j i σj~P , and apply a Bogoliubov transformation by conjugating withthe unitary operator W σ j +1 ( ~ ∇ E σ j ~P ). Formally, we find that b K σ j +1 ~P = (cid:0) ~ Γ σ j ~P + ~ L σ j σ j +1 + ~ I σ j σ j +1 (cid:1) X λ Z R | ~k | δ σ j ~P ( b k ) b ∗ ~k,λ b ~k,λ d k + b E σ j +1 ~P where ~ L σ j σ j +1 := − α X λ Z B σj \B σj +1 ~k ~ ∇ E σ j ~P · ~ε ∗ ~k,λ b ~k,λ + h.c. | ~k | δ σ j ~P ( b k ) d k − α ~A | σ j σ j +1 (IV.50) ~ I σ j σ j +1 := α X λ Z B σj \B σj +1 ~k ~ ∇ E σ j ~P · ~ε ∗ ~k,λ ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | ( δ σ j ~P ( b k )) d k (IV.51)+ α X λ Z B σj \B σj +1 (cid:2) ~ε ~k,λ ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) + h.c. (cid:3) d k q | ~k | b E σ j +1 ~P := ~P − ( ~P − ~ ∇ E σ j ~P ) − α X λ Z B Λ \B σj +1 | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) d k . For details on the derivation of (IV.49) and for the proof that (IV.40) and (IV.49) hold in theoperator sense (and not only formally), we refer to Lemmata A.1 and A.2 in the Appendix.We also define the operators ( j ≥ b ~ Π σ j ~P := W σ j ( ~ ∇ E σ j − ~P ) W ∗ σ j ( ~ ∇ E σ j ~P ) ~ Π σ j ~P W σ j ( ~ ∇ E σ j ~P ) W ∗ σ j ( ~ ∇ E σ j − ~P ) (IV.53) FP Infraparticle Scattering b ~ Γ σ j ~P := b ~ Π σ j ~P − h b ~ Π σ j ~P i b Φ σj~P , (IV.54)which are used in the proofs in the next section. Here, b Φ σ j ~P denotes the ground state vectorof the Hamiltonian b K σ j ~P := W σ j ( ~ ∇ E σ j − ~P ) H σ j ~P W ∗ σ j ( ~ ∇ E σ j − ~P ). C. Construction and convergence of { Φ σ j ~P } In this section, we construct a sequence { Φ σ j ~P | j ∈ N } of unnormalized ground statevectors of the (Bogoliubov-transformed) Hamiltonians K σ j ~P , and establish the existence of s − lim j →∞ Φ σ j ~P . (IV.55)(We warn the reader that, with an abuse of notation, we use the same symbol introducedfor the normalized ground state vector in (III.8).)In the initial step of the construction corresponding to j = 0, we define Φ σ ~P := Ω f , with k Ω f k = 1.To pass from scale j to j + 1, we proceed in two steps. First, we construct an intermediatevector b Φ σ j +1 ~P b Φ σ j +1 ~P = ∞ X n =0 πi Z γ j +1 dz j +1 K σ j ~P − z j +1 (cid:2) − ∆ K ~P | σ j σ j +1 K σ j ~P − z j +1 (cid:3) n Φ σ j ~P , (IV.56)where ∆ K ~P | σ j σ j +1 := b K σ j +1 ~P − b E σ j +1 ~P + E σ j ~P − K σ j ~P = 12 h ~ Γ σ j ~P · (cid:0) ~ L σ j σ j +1 + ~ I σ j σ j +1 (cid:1) + h.c. i + (cid:0) ~ L σ j σ j +1 + ~ I σ j σ j +1 (cid:1) . (IV.57)Then, we define Φ σ j +1 ~P := W σ j +1 ( ~ ∇ E σ j +1 ~P ) W ∗ σ j +1 ( ~ ∇ E σ j ~P ) b Φ σ j +1 ~P . (IV.58)The series in (IV.56) is termwise well-defined and converges strongly to a non-zero vector,provided α is small enough (independently of j ). This follows from operator-norm estimatesof the type used for (IV.18).To prove the convergence of the sequence { Φ σ j ~P } , we proceed as follows. The key point isto show that the term 12 h ~ Γ σ j ~P · (cid:0) ~ L σ j σ j +1 + ~ I σ j σ j +1 (cid:1) + h.c. i (IV.59) FP Infraparticle Scattering (cid:10) Φ σ j ~P , ~ Γ σ j ~P Φ σ j ~P (cid:11) = 0 , (IV.60)as we will show. We then proceed to showing that terms like (cid:13)(cid:13)(cid:0) K σ j ~P − z j +1 (cid:1) (cid:2) ~ Γ σ j ~P · (cid:0) ~ L σ j (+) σ j +1 + ~ I σ j σ j +1 (cid:1)(cid:3)(cid:0) K σ j ~P − z j +1 (cid:1) Φ σ j ~P (cid:13)(cid:13) (IV.61)(where ~ L σ j (+) σ j +1 stands for the part which contains only photon creation operators) are oforder O ( ǫ ηj ), for some η >
0, and we consequently deduce that k b Φ σ j +1 ~P − Φ σ j ~P k (IV.62)tends to 0, as j → ∞ . Theorem IV.1.
The strong limit s − lim j →∞ Φ σ j ~P (IV.63) exists and is non-zero, and the rate of convergence is, at least, O ( σ (1 − δ ) j ) , for any < δ < . In the proof, we can import results from [22] at various places. Thus, we will be sketchyin part of our presentation.
D. Key ingredients of the proof of Theorem IV.1 • Constraints on ǫ , µ and α In addition to the conditions on α , ǫ and µ imposed in our discussion so far, the analysisin this part will require an upper bound on µ and an upper bound on ǫ strictly smaller thanthe ones imposed by the inequalities (IV.8), (IV.9); see Lemma A.3 and (IV.90) below.We note that the more restrictive conditions on µ and ǫ imply new bounds on ρ − , ρ + .Moreover, ǫ must satisfy a lower bound ǫ > C α , with C > • Estimates on the shift of the ground state energy and its gradient
There are constants C , C such that the following hold. FP Infraparticle Scattering A | E σ j ~P − E σ j +1 ~P | ≤ C α ǫ j (IV.64)This estimate can be proved as inequality (II.19) in [4].( A | ~ ∇ E σ j +1 ~P − ~ ∇ E σ j ~P | ≤ C (cid:16)(cid:13)(cid:13)(cid:13) b Φ σ j +1 ~P k b Φ σ j +1 ~P k − Φ σ j ~P k Φ σ j ~P k (cid:13)(cid:13)(cid:13) + ǫ j +12 (cid:17) (IV.65)For the proof, see Lemma A.2 in the Appendix. • Bounds relating expectations of operators to those of their absolute values
There are constants C , C > A
3) For z j +1 ∈ γ j +1 , D (Γ σ j ~P ) i Φ σ j ~P , (cid:12)(cid:12) K σ j ~P − z j +1 (cid:12)(cid:12) (Γ σ j ~P ) i Φ σ j ~P E ≤ C (cid:12)(cid:12)(cid:12)D (Γ σ j ~P ) i Φ σ j ~P , K σ j ~P − z j +1 (Γ σ j ~P ) i Φ σ j ~P E(cid:12)(cid:12)(cid:12) . (IV.66)( A
4) For z j +1 ∈ γ j +1 , D ~ L σ j (+) σ j +1 (Γ σ j ~P ) i Φ σ j ~P , (cid:12)(cid:12) K σ j ~P − z j +1 (cid:12)(cid:12) ~ L σ j (+) σ j +1 (Γ σ j ~P ) i Φ σ j ~P E ≤ C (cid:12)(cid:12)(cid:12)D ~ L σ j (+) σ j +1 (Γ σ j ~P ) i Φ σ j ~P , K σ j ~P − z j +1 ~ L σ j (+) σ j +1 (Γ σ j ~P ) i Φ σ j ~P E(cid:12)(cid:12)(cid:12) . (IV.67)To obtain these two bounds, it suffices to exploit the fact that the spectral support(with respect to K σ j ~P ) of the two vectors (Γ σ j ~P ) i Φ σ j ~P and ~ L σ j (+) σ j +1 (Γ σ j ~P ) i Φ σ j ~P is strictly abovethe ground state energy, since they are both orthogonal to the ground state Φ σ j ~P . Remark:
The constants C , . . . , C are independent of α , ǫ , µ , and j ∈ N , provided that α, ǫ , and µ are sufficiently small. FP Infraparticle Scattering E. Proof of the convergence of (Φ σ j ~P ) ∞ j =0 The proof of Theorem IV.1 consists of four main steps.
Step (1) (i) Assuming the bound (cid:12)(cid:12)(cid:12)D (Γ σ j ~P ) i Φ σ j ~P , K σ j ~P − z j +1 (Γ σ j ~P ) i Φ σ j ~P E(cid:12)(cid:12)(cid:12) ≤ R α ǫ jδ > δ > , (IV.68) where R is a constant uniform in j ∈ N , for α , ǫ , µ sufficiently small, we prove that (cid:13)(cid:13)(cid:0) K σ j ~P − z j +1 (cid:1) (cid:2) ~ Γ σ j ~P · (cid:0) ~ L σ j (+) σ j +1 + ~ I σ j σ j +1 (cid:1)(cid:3)(cid:0) K σ j ~P − z j +1 (cid:1) Φ σ j ~P (cid:13)(cid:13) ≤ O ( ǫ j (1 − δ ) ) ; (IV.69) (see (IV.61)). (ii) For α and R small enough independently of j , we prove that k b Φ σ j +1 ~P − Φ σ j ~P k ≤ ǫ j +12 (1 − δ ) . (IV.70)For the term on the l.h.s. of (IV.69) proportional to to ~ I σ j σ j +1 , the asserted upper boundis readily obtained from estimate ( A
3) combined with (IV.68). For the term proportionalto ~ L σ j (+) σ j +1 , we prove (IV.69) following arguments developed in [22]; see Lemma A.3 of theAppendix for details. This involves the application of a ”pull-through formula”, a resolventexpansion, and the bounds ( A A Step (2)
We relate the l.h.s. of (IV.68) to the corresponding quantity with j replaced by j − , andto the norm difference k b Φ σ j ~P − Φ σ j − ~P k (IV.71) (see (IV.80) – (IV.83) below). FP Infraparticle Scattering W σ j ( ~ ∇ E σ j − ~P ) W ∗ σ j ( ~ ∇ E σ j ~P ), the l.h.s. of (IV.68) equals (cid:12)(cid:12)(cid:12)D ( b Γ σ j ~P ) i b Φ σ j ~P , b K σ j ~P − z j +1 ( b Γ σ j ~P ) i b Φ σ j ~P E(cid:12)(cid:12)(cid:12) . (IV.72)Assuming that α is small enough and ǫ > C α , with C > A
1) to re-expand the resolvent and find (cid:12)(cid:12)(cid:12)D ( b Γ σ j ~P ) i b Φ σ j ~P , b K σ j ~P − z j +1 ( b Γ σ j ~P ) i b Φ σ j ~P E(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)D ( b Γ σ j ~P ) i b Φ σ j ~P , (cid:12)(cid:12)(cid:12) K σ j − ~P − z j +1 (cid:12)(cid:12)(cid:12) ( b Γ σ j ~P ) i b Φ σ j ~P E(cid:12)(cid:12)(cid:12) . (IV.73)We then readily obtain that2 (cid:12)(cid:12)(cid:12)D ( b Γ σ j ~P ) i b Φ σ j ~P , (cid:12)(cid:12)(cid:12) K σ j − ~P − z j +1 (cid:12)(cid:12)(cid:12) ( b Γ σ j ~P ) i b Φ σ j ~P E(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:16) K σ j − ~P − z j +1 (cid:17) (( b Γ σ j ~P ) i b Φ σ j ~P − (Γ σ j − ~P ) i Φ σ j − ~P ) (cid:13)(cid:13)(cid:13) (IV.74)+ 4 (cid:12)(cid:12)(cid:12)D (Γ σ j − ~P ) i Φ σ j − ~P , (cid:12)(cid:12)(cid:12) K σ j − ~P − z j +1 (cid:12)(cid:12)(cid:12) (Γ σ j − ~P ) i Φ σ j − ~P E(cid:12)(cid:12)(cid:12) . (IV.75)Our strategy is to construct a recursion that relates (IV.75) to the initial expression (IV.72)with j replaced by j −
1, while (IV.74) is a remainder term.We bound the remainder term (IV.74) by4 (cid:13)(cid:13)(cid:13)(cid:16) K σ j − ~P − z j +1 (cid:17) (( b Γ σ j ~P ) i b Φ σ j ~P − (Γ σ j − ~P ) i Φ σ j − ~P ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:16) K σ j − ~P − z j +1 (cid:17) (( b Γ σ j ~P ) i b Φ σ j ~P − (Γ σ j − ~P ) i b Φ σ j ~P ) (cid:13)(cid:13)(cid:13) (IV.76)+ 8 (cid:13)(cid:13)(cid:13)(cid:16) K σ j − ~P − z j +1 (cid:17) (Γ σ j − ~P ) i ( b Φ σ j ~P − Φ σ j − ~P ) (cid:13)(cid:13)(cid:13) ≤ R ǫ j (cid:16) k b Φ σ j ~P − Φ σ j − ~P k + ǫ j ǫ j (cid:17) (IV.77)+ R ǫ j (cid:16) (cid:13)(cid:13) b Φ σj~P k b Φ σj~P k − Φ σj − ~P k Φ σj − ~P k (cid:13)(cid:13) + ǫ j ǫ j (cid:17) , where R and R are constants independent of α , µ , and j ∈ N , provided that α , ǫ , and µ are sufficiently small, and ǫ > C α . For details on the step from (IV.76) to (IV.77), werefer to Lemma A.4 of the Appendix. FP Infraparticle Scattering A
3) and the orthogonality property expressed in(IV.60). We find that, for any z j ∈ γ j ,4 (cid:12)(cid:12)(cid:12)D (Γ σ j − ~P ) i Φ σ j − ~P , (cid:12)(cid:12)(cid:12) K σ j − ~P − z j +1 (cid:12)(cid:12)(cid:12) (Γ σ j − ~P ) i Φ σ j − ~P E(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)D (Γ σ j − ~P ) i Φ σ j − ~P , K σ j − ~P − z j +1 (Γ σ j − ~P ) i Φ σ j − ~P E(cid:12)(cid:12)(cid:12) (IV.78) ≤ C (cid:12)(cid:12)(cid:12)D (Γ σ j − ~P ) i Φ σ j − ~P , K σ j − ~P − z j (Γ σ j − ~P ) i Φ σ j − ~P E(cid:12)(cid:12)(cid:12) . (IV.79)In passing from (IV.78) to (IV.79), we have used the constraint on the spectral support(with respect to K σ j − ~P ) of the vector (Γ σ j − ~P ) i Φ σ j − ~P .Therefore, for sufficiently small values of the parameters ǫ and α , we conclude that (cid:12)(cid:12)(cid:12)D (Γ σ j ~P ) i Φ σ j ~P , K σ j ~P − z j +1 (Γ σ j ~P ) i Φ σ j ~P E(cid:12)(cid:12)(cid:12) (IV.80) ≤ R ǫ j (cid:16) k b Φ σ j ~P − Φ σ j − ~P k + ǫ j ǫ j (cid:17) (IV.81)+ R ǫ j (cid:16) (cid:13)(cid:13) b Φ σj~P k b Φ σj~P k − Φ σj − ~P k Φ σj − ~P k (cid:13)(cid:13) + ǫ j ǫ j (cid:17) (IV.82)+ 8 C (cid:12)(cid:12)(cid:12)D (Γ σ j − ~P ) i Φ σ j − ~P , K σ j − ~P − z j (Γ σ j − ~P ) i Φ σ j − ~P E(cid:12)(cid:12)(cid:12) . (IV.83) Step (3)
We prove that k Φ σ j ~P − b Φ σ j ~P k ≤ C α | ~ ∇ E σ j − ~P − ~ ∇ E σ j ~P | | ln( ǫ j ) | , (IV.84)where C is independent of α , ǫ , µ , and j ∈ N , provided that α , ǫ , and µ are sufficientlysmall.¿From the definition Φ σ j ~P := W σ j ( ~ ∇ E σ j ~P ) W ∗ σ j ( ~ ∇ E σ j − ~P ) b Φ σ j ~P , (IV.85)we get that k Φ σ j ~P − b Φ σ j ~P k = k W ∗ σ j ( ~ ∇ E σ j − ~P ) W σ j ( ~ ∇ E σ j ~P )Ψ σ j ~P − Ψ σ j ~P k (IV.86) FP Infraparticle Scattering σ j ~P the ground state eigenvector W ∗ σ j ( ~ ∇ E σ j ~P )Φ σ j ~P , (IV.87) k W ∗ σ j ( ~ ∇ E σ j ~P )Φ σ j ~P k ≤
1, of the Hamiltonian H σ j ~P . Then, we apply formula (III.16) (which wasderived in [9]), and obtain the logarithmic bound h N f i Ψ σj~P ≤ O ( | ln σ j | ) for the expectationvalue of the photon number operator N f in Ψ σ j ~P , where σ j = Λ ǫ j , and Λ ≈
1. Hence, theestimate (IV.84) follows.
Step (4)
We prove the bound (IV.68) assumed in step (1) by an inductive argument (see (IV.95)below) .We assume α , ǫ , and µ to be sufficiently small for all our previous results to hold, andsuch that:i) S j := j X m =1 h ǫ m (1 − δ ) + 4 C C α ǫ m (1 − δ ) | ln( ǫ m ) | i ≤ , (IV.88)uniformly in j .ii) k b Φ σ ~P − Φ σ ~P k ≤ ǫ (1 − δ ) . (IV.89)iii) The bound (IV.68) holds for j = 1, and0 < R + R ≤ (1 − C ǫ δ ) R α . (IV.90)Notably, (IV.90) imposes a more restrictive upper bound on the admissible values of ǫ .Then, we proceed with the induction in j . • Inductive hypotheses
We assume that, for j − ≥ FP Infraparticle Scattering H
1) we have an estimate k Φ σ j − ~P − Φ σ ~P k ≤ S j − = j − X m =1 h ǫ m (1 − δ ) + 4 C C α ǫ m (1 − δ ) | ln( ǫ m ) | i ;( H
2) the bound (IV.68) holds for j − ≥ • Induction step from j − to j ¿From ( H k b Φ σ j ~P − Φ σ j − ~P k ≤ ǫ j (1 − δ ) . (IV.91)¿From ( H H
2) and ( A k Φ σ j − ~P k ≥ k Φ σ ~P k − k Φ σ j − ~P − Φ σ ~P k ≥
23 (IV.92) | ~ ∇ E σ j ( ~P ) − ~ ∇ E σ j − ( ~P ) | ≤ C ǫ j (1 − δ ) . (IV.93)and then, by combining (IV.91), (IV.84) and (IV.65), that k Φ σ j ~P − Φ σ ~P k ≤ S j . (IV.94)Finally, we obtain from (IV.81) – (IV.83) that (cid:12)(cid:12)(cid:12)D (Γ σ j ~P ) i Φ σ j ~P , K σ j ~P − z j +1 (Γ σ j ~P ) i Φ σ j ~P E(cid:12)(cid:12)(cid:12) ≤ R ǫ j (cid:16) k b Φ σ j ~P − Φ σ j − ~P k + ǫ j (1 − δ ) ǫ j (cid:17) + R ǫ j (cid:16) (cid:13)(cid:13) b Φ σj~P k b Φ σj~P k − Φ σj − ~P k Φ σj − ~P k (cid:13)(cid:13) + ǫ j (1 − δ ) ǫ j (cid:17) + 8 C (cid:12)(cid:12)(cid:12)D (Γ σ j − ~P ) i Φ σ j − ~P , K σ j − ~P − z j (Γ σ j − ~P ) i Φ σ j − ~P E(cid:12)(cid:12)(cid:12) ≤ R ǫ jδ + R ǫ jδ + 8 C R αǫ ( j − δ ≤ R αǫ jδ . (IV.95)This proves (IV.69) and implies that the sequence { Φ σ j ~P } converges. This follows from thesame argument yielding (IV.94). The limit is a non-zero vector because of (IV.92) whichholds uniformly in j .This concludes the proof of statement ( I
1) in Theorem III.1 for the sequence σ j = Λ ǫ j of infrared cutoffs. For general sequences of infrared cutoffs, ( I
1) follows by arguments in[22].
FP Infraparticle Scattering V. PROOF OF STATEMENTS ( I AND ( I IN THE MAIN THEOREM
Statement ( I
2) in Theorem III.1 expresses H¨older regularity of Φ σ~P and ~ ∇ E σ~P with respectto ~P ∈ S , uniformly in σ ≥
0. That is, k Φ σ~P − Φ σ~P +∆ ~P k ≤ C δ ′ | ∆ ~P | − δ ′ (V.1)and | ~ ∇ E σ~P − ~ ∇ E σ~P +∆ ~P | ≤ C δ ′′ | ∆ ~P | − δ ′′ , (V.2)for any 0 < δ ′′ < δ ′ < , where ~P , ~P + ∆ ~P ∈ S . The constants C δ ′ and C δ ′′ depend on δ ′ and δ ′′ , respectively. This result can be taken over from [22].Statement ( I
3) follows easily from ( I ~P − ~ ∇ E σ~P = (cid:10) ~P f − α / ~A σ (cid:11) Ψ σ~P . (V.3)We then find that | (cid:10) ~P f (cid:11) Ψ σ~P | ≤ X λ Z d k | ~k | k b ~k,λ Ψ σ~P k k Ψ σ~P k ≤ C ′ α Z B Λ d k | ~k | ≤ C α , (V.4)and | (cid:10) α / ~A σ (cid:11) Ψ σ~P | ≤ α / X λ Z d k | ~k | / k b ~k,λ Ψ σ~P kk Ψ σ~P k≤ C ′ α Z B Λ d k | ~k | ≤ C α , (V.5)where we used ( I
5) in (V.5). Therefore, | ~P − ~ ∇ E σ~P | ≤ C α , (V.6)for a constant C independent of ~P ∈ S and σ . Statement ( I
3) then follows immediately.
FP Infraparticle Scattering VI. PROOF OF STATEMENT ( I IN THE MAIN THEOREM
To prove statement ( I
4) in Theorem III.1, we must show that, for ~P ∈ S , α smallenough, ~k = 0 and σ ≥ E σ~P − ~k > E σ~P − C α | ~k | (VI.1)holds, where E σ~P − ~k := inf spec H σ~P − ~k , and < C α <
1, with C α → as α → H σ~P + ~k = H σ~P + ~k · ~ ∇ H σ~P + | ~k | , (VI.2)and that (cid:10) φ , H σ~P + ~k φ (cid:11) ≥ (cid:10) φ , H σ~P φ (cid:11) − | ~k | (cid:10) φ , ( ~ ∇ H σ~P ) φ (cid:11) / + | ~k | ≥ (cid:10) φ , H σ~P φ (cid:11) − √ | ~k | (cid:10) φ , H σ~P φ (cid:11) / + | ~k | φ ∈ D ( H σ~P + ~k ), with k φ k = 1. Thus, we obtain the inequality (cid:10) φ , H σ~P + ~k φ (cid:11) − E σ~P ≥ inf z ≥ (cid:8) ( z + E σ~P ) − √ | ~k | ( z + E σ~P ) / + | ~k | − E σ~P (cid:9) = inf x ≥ ( E σ~P ) / (cid:8) x − √ | ~k | x + | ~k | − E σ~P (cid:9) , (VI.4)where z := (cid:10) φ , H σ~P φ (cid:11) − E σ~P ≥ ∂ x ( · · · ) = 0 in the expression on the last line of (VI.4), we find2 x − √ | ~k | = 0 . (VI.5)The minimum is therefore attained at x = √ | ~k | , if √ | ~k | ≥ ( E σ~P ) / , and at x = ( E σ~P ) / ,corresponding to z = 0, otherwise. That is, x min = max { √ | ~k | , ( E σ~P ) / } . (VI.6)Now, for √ | ~k | ≥ ( E σ~P ) / , so that x min = √ | ~k | , we evaluate the lower bound and get | ~k | − | ~k | + | ~k | − E σ~P , (VI.7)and we observe that − E σ~P ≥ − | ~k | , (VI.8) FP Infraparticle Scattering E σ~P ≤ √ cα ) ( E σ~P ) / ≤ | ~k | (VI.9)for | ~P | < and α small enough. This follows from0 < E σ~P = infspec H σ~P ≤ (cid:10) Ω f , H σ~P Ω f (cid:11) = 12 | ~P | + α (cid:10) ( ~A σ ) (cid:11) (VI.10)by Rayleigh-Ritz, so that ( E σ~P ) / ≤ √ ( + cα ) for | ~P | < and α small enough.If, however, √ | ~k | ≤ ( E σ~P ) / , so that x min = ( E σ~P ) / , evaluation of the lower bound yields − √ | ~k | ( E σ~P ) / + | ~k | , (VI.11)and we observe that − √ | ~k | ( E σ~P ) / + | ~k | ≥ − ( | ~P | + cα ) | ~k | ≥ − ( 13 + cα ) | ~k | (VI.12)for | ~P | < .Therefore, we conclude that E σ~P + ~k > E σ~P − C α | ~k | (VI.13)for C α = 13 + cα , (VI.14)and all ~k = 0.This establishes statement ( I
4) in Theorem III.1.Thus, we have proven our main result, up to auxiliary results proven in the Appendix.
FP Infraparticle Scattering APPENDIX A:1. Well-definedness of the operators K σ j ~P and b K σ j ~P We need to verify that the canonical form of the Hamiltonians K σ j ~P and b K σ j ~P in (IV.40)and (IV.49) are not only formal. This can be achieved by adapting an argument in the work[21] of E. Nelson, Lemma 3. We shall only outline the proof for K σ j ~P ; the case of b K σ j ~P issimilar.To this end, we write ( K σ j ~P ) ′ for the operator on the right hand side of (IV.40), in orderto distinguish it from (IV.38). We let H ~P ( ∞ ) denote the linear span of vectors in H ~P witha finite number of photons. For the values of α and of Λ assumed in Section II, we knowthat H σ j ~P is selfadjoint in D ( H ~P ), where H ~P := ( ~P − ~P f ) H f . (A.1)Then, we conclude the following:1) The equality (IV.40) trivially holds on H ~P ( ∞ ) T D ( H ~P ), because vectors in this spaceare analytic for the generator of W σ j ( ~ ∇ E σ j ~P ), and since H σ j ~P , H ~P and the generator of W σ j ( ~ ∇ E σ j ~P ) map H ~P ( ∞ ) T D ( H ~P ) into itself.2) By standard arguments, one shows that k H ~P W σ j ( ~ ∇ E σ j ~P ) ψ k ≤ b ( k H ~P ψ k + k ψ k ) , (A.2)where ψ ∈ H ~P ( ∞ ) T D ( H ~P ), for some b > H ~P ( ∞ ) T D ( H ~P ) is dense in D ( H ~P ) with respect to the norm k H ~P ψ k + k ψ k ,it follows that W σ j ( ~ ∇ E σ j ~P ) and W ∗ σ j ( ~ ∇ E σ j ~P ) map D ( H ~P ) into itself.Consequently, D ( H ~P ) ≡ D ( K σ j ~P ) . (A.3)3) The equality (IV.40) holds on D ( K σ j ~P ) because H ~P ( ∞ ) T D ( H ~P ) is dense in D ( H ~P )in the norm k H ~P ψ k + k ψ k , and because of (A.3). Since ( K σ j ~P ) ′ ≡ K σ j ~P on the do-main of selfadjointness of K σ j ~P , we can therefore conclude that D (( K σ j ~P ) ′ ) ≡ D ( K σ j ~P ).Consequently, we have proven that ( K σ j ~P ) ′ ≡ K σ j ~P . This is what we intended to prove. FP Infraparticle Scattering
2. Technical lemmata for the proof of ( I in Theorem III.1 Lemma A.1.
The Hamiltonian b K σ j +1 ~P has the form (IV.49), with (IV.50), (IV.51), and(IV.52).Proof. Recalling the definitions of Section IV B 1, we have W σ j +1 ( ~ ∇ E σ j ~P ) ~β σ j +1 W ∗ σ j +1 ( ~ ∇ E σ j ~P ) − h ~β σ j i Ψ σj~P (A.4)= W σ j +1 ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j +1 ( ~ ∇ E σ j ~P ) − h ~β σ j i Ψ σj~P (A.5) − α W σ j +1 ( ~ ∇ E σ j ~P ) ~A σ j σ j +1 W ∗ σ j +1 ( ~ ∇ E σ j ~P ) (A.6)= W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) − h ~β σ j i Ψ σj~P (A.7)+ W σ j σ j +1 ( ~ ∇ E σ j ~P ) ~P f W σ j ∗ σ j +1 ( ~ ∇ E σ j ~P ) − ~P f (A.8) − α W σ j +1 ( ~ ∇ E σ j ~P ) ~A σ j σ j +1 W ∗ σ j +1 ( ~ ∇ E σ j ~P ) (A.9)= ~ Π σ j ~P − h ~ Π σ j ~P i Φ σj~P (A.10) − α X λ Z B σj \B σj +1 ~k ~ ∇ E σ j ~P · ~ε ∗ ~k,λ b ~k,λ + h.c. | ~k | δ σ j ~P ( b k ) d k − α ~A σ j σ j +1 (A.11)+ α X λ Z B σj \B σj +1 ~k ~ ∇ E σ j ~P · ~ε ∗ ~k,λ ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | ( δ σ j ~P ( b k )) d k (A.12)+ α X λ Z B σj \B σj +1 (cid:2) ~ε ~k,λ ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) + h.c. (cid:3) d k q | ~k | , (A.13)where W σ j σ j +1 ( ~ ∇ E σ j ~P ) := exp (cid:16) α X λ Z B σj \B σj +1 d k ~ ∇ E σ j ~P | ~k | δ σ j ~P ( b k ) · ( ~ε ~k,λ b ∗ ~k,λ − h.c. ) (cid:17) . (A.14)This establishes (IV.50) and (IV.51). FP Infraparticle Scattering Lemma A.2.
For ~P ∈ S , there exists C > such that, uniformly in j ∈ N , the inequality | ~ ∇ E σ j +1 ~P − ~ ∇ E σ j ~P | ≤ C (cid:16)(cid:13)(cid:13)(cid:13) b Φ σ j +1 ~P k b Φ σ j +1 ~P k − Φ σ j ~P k Φ σ j ~P k (cid:13)(cid:13)(cid:13) + ǫ j +12 (cid:17) (A.15) holds.Proof. Using (IV.37) and (IV.53), we write ~ ∇ E σ j +1 ~P and ~ ∇ E σ j ~P in the form ~ ∇ E σ j ~P = ~P − (cid:10) Φ σ j ~P , ~ Π σ j ~P Φ σ j ~P (cid:11)(cid:10) Φ σ j ~P , Φ σ j ~P (cid:11) − h W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) i Ω f (A.16) ~ ∇ E σ j +1 ~P = ~P − (cid:10) b Φ σ j +1 ~P , b ~ Π σ j +1 ~P b Φ σ j +1 ~P (cid:11)(cid:10) b Φ σ j +1 ~P , b Φ σ j +1 ~P (cid:11) (A.17) −h W σ j +1 ( ~ ∇ E σ j +1 ~P ) ~β σ j +1 W ∗ σ j +1 ( ~ ∇ E σ j +1 ~P ) i Ω f . By a simple, but slightly lengthy calculation, one can check that h W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) i Ω f − (A.18) −h W σ j +1 ( ~ ∇ E σ j +1 ~P ) ~β σ j +1 W ∗ σ j +1 ( ~ ∇ E σ j +1 ~P ) i Ω f (A.19)= α X λ Z Λ \B σj ~k ~ ∇ E σ j ~P · ~ε ∗ ~k,λ ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | ( δ σ j ~P ( b k )) d k (A.20) − α X λ Z Λ \B σj ~k ~ ∇ E σ j +1 ~P · ~ε ∗ ~k,λ ~ ∇ E σ j +1 ~P · ~ε ~k,λ | ~k | ( δ σ j +1 ~P ( b k )) d k (A.21)+ α X λ Z Λ \B σj (cid:2) ~ε ~k,λ ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) + h.c. (cid:3) d k q | ~k | (A.22) − α X λ Z Λ \B σj (cid:2) ~ε ~k,λ ~ ∇ E σ j +1 ~P · ~ε ∗ ~k,λ | ~k | δ σ j +1 ~P ( b k ) + h.c. (cid:3) d k q | ~k | (A.23) − α X λ Z B σj \B σj +1 ~k ~ ∇ E σ j +1 ~P · ~ε ∗ ~k,λ ~ ∇ E σ j +1 ~P · ~ε ~k,λ | ~k | ( δ σ j +1 ~P ( b k )) d k (A.24) − α X λ Z B σj \B σj +1 (cid:2) ~ε ~k,λ ~ ∇ E σ j +1 ~P · ~ε ∗ ~k,λ | ~k | δ σ j +1 ~P ( b k ) + h.c. (cid:3) d k q | ~k | . (A.25) FP Infraparticle Scattering b ~ Π σ j +1 ~P − ~ Π σ j ~P (A.26)= ~ L σ j σ j +1 (A.27)+ α X λ Z Λ \B σj +1 ~k ~ ∇ E σ j ~P · ~ε ∗ ~k,λ ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | ( δ σ j ~P ( b k )) d k (A.28) − α X λ Z Λ \B σj +1 ~k ~ ∇ E σ j +1 ~P · ~ε ∗ ~k,λ ~ ∇ E σ j +1 ~P · ~ε ~k,λ | ~k | ( δ σ j +1 ~P ( b k )) d k (A.29)+ α X λ Z Λ \B σj +1 (cid:2) ~ε ~k,λ ~ ∇ E σ j ~P · ~ε ∗ ~k,λ | ~k | δ σ j ~P ( b k ) + h.c. (cid:3) d k q | ~k | (A.30) − α X λ Z Λ \B σj +1 (cid:2) ~ε ~k,λ ~ ∇ E σ j +1 ~P · ~ε ∗ ~k,λ | ~k | δ σ j +1 ~P ( b k ) + h.c. (cid:3) d k q | ~k | . (A.31)In order to shorten our notations, we define F j := (A.20) + (A.21) + (A.22) + (A.23) (A.32) F j +1 := (A.28) + (A.29) + (A.30) + (A.31) (A.33) G jj +1 := (A.24) + (A.25) (A.34)Returning to (A.16), (A.18), we can write ~ ∇ E σ j +1 ~P − ~ ∇ E σ j ~P − F j (A.35)= − k b Φ σ j +1 ~P k D b Φ σ j +1 ~P , b ~ Π σ j +1 ~P ( b Φ σ j +1 ~P k b Φ σ j +1 ~P k − Φ σ j ~P k Φ σ j ~P k ) E (A.36) − (cid:10) b Φ σ j +1 ~P , b ~ Π σ j +1 ~P Φ σ j ~P (cid:11) k b Φ σ j +1 ~P k k Φ σ j ~P k + (cid:10) b Φ σ j +1 ~P , ~ Π σ j ~P Φ σ j ~P (cid:11) k b Φ σ j +1 ~P k k Φ σ j ~P k (A.37) − (cid:10) b Φ σ j +1 ~P , ~ Π σ j ~P Φ σ j ~P (cid:11) k b Φ σ j +1 ~P k k Φ σ j ~P k + (cid:10) Φ σ j ~P , ~ Π σ j ~P Φ σ j ~P (cid:11) k Φ σ j ~P k + G jj +1 . (A.38) FP Infraparticle Scattering ~ ∇ E σ j +1 ~P − ~ ∇ E σ j ~P − F j + (cid:10) b Φ σ j +1 ~P , Φ σ j ~P (cid:11) k b Φ σ j +1 ~P k k Φ σ j ~P k F j +1 (A.39)= − k b Φ σ j +1 ~P k D b Φ σ j +1 ~P , b ~ Π σ j +1 ~P (cid:0) b Φ σ j +1 ~P k b Φ σ j +1 ~P k − Φ σ j ~P k Φ σ j ~P k (cid:1) E (A.40) − k Φ σ j ~P k D (cid:0) b Φ σ j +1 ~P k b Φ σ j +1 ~P k − Φ σ j ~P k Φ σ j ~P k (cid:1) , ~ Π σ j ~P Φ σ j ~P E (A.41) − (cid:10) b Φ σ j +1 ~P , ~ L σ j σ j +1 Φ σ j ~P (cid:11) k b Φ σ j +1 ~P k k Φ σ j ~P k + G jj +1 . (A.42)We deduce from the definitions (A.32) and (A.33) that | F j | , | F j +1 | < c ′ | ~ ∇ E σ j +1 ~P − ~ ∇ E σ j ~P | (A.43)where c ′ is O ( α ) and j -independent. Then, it suffices to check that, for α small enough,there are positive constants c , C uniform in j , such that C (cid:16)(cid:13)(cid:13)(cid:13) b Φ σ j +1 ~P k b Φ σ j +1 ~P k − Φ σ j ~P k Φ σ j ~P k (cid:13)(cid:13)(cid:13) + ǫ j +12 (cid:17) (A.44) ≥ (cid:12)(cid:12) (A.40) + (A.41) + (A.42) (cid:12)(cid:12) ≥ c | ~ ∇ E σ j +1 ~P − ~ ∇ E σ j ~P | (A.45)is satisfied. FP Infraparticle Scattering Lemma A.3.
Assume ~P ∈ S , and α , µ , and ǫ small enough. Then, uniformly in j ∈ N ,the bound (cid:13)(cid:13)(cid:0) K σ j ~P − z j +1 (cid:1) L σ j (+) lσ j +1 (Γ σ j ~P ) l (cid:0) K σ j ~P − z j +1 (cid:1) Φ σ j ~P (cid:13)(cid:13) (A.46) ≤ − c C C Z jj +1 | E σ j − ~P − z j +1 | (cid:12)(cid:12)(cid:12) D (Γ σ j ~P ) l Φ σ j ~P , K σ j ~P − z j +1 (Γ σ j ~P ) l Φ σ j ~P E(cid:12)(cid:12)(cid:12) holds for each l = 1 , , , where γ σ j +1 := { z j +1 ∈ C (cid:12)(cid:12) | z j +1 − E σ j ~P | = µσ j +1 } , and c < . C and C are defined in (IV.66), (IV.67) ( ( A and ( A from Section IV D), and Z jj +1 := hL σ j ( − ) lσ j +1 L σ j (+) lσ j +1 i Ω f (A.47)= α X λ Z B σj \B σj +1 d k (cid:12)(cid:12)(cid:12) k l ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) + ( b l · ~ε ~k,λ ) q | ~k | ) (cid:12)(cid:12)(cid:12) . Proof.
We first use Eq. (IV.67) to estimate (cid:13)(cid:13)(cid:0) K σ j ~P − z j +1 (cid:1) L σ j (+) lσ j +1 (Γ σ j ~P ) l Φ σ j ~P (cid:13)(cid:13) (A.48) ≤ D L σ j (+) lσ j +1 (Γ σ j ~P ) l Φ σ j ~P , (cid:12)(cid:12) K σ j ~P − z j +1 (cid:12)(cid:12) L σ j (+) lσ j +1 (Γ σ j ~P ) l Φ σ j ~P E ≤ C (cid:12)(cid:12)(cid:12)D L σ j (+) lσ j +1 (Γ σ j ~P ) l Φ σ j ~P , K σ j ~P − z j +1 L σ j (+) lσ j +1 (Γ σ j ~P ) l Φ σ j ~P E(cid:12)(cid:12)(cid:12) . (A.49)Then we use pull-through formula to derive the following equality which holds in the senseof distributions for ~k ∈ B σ j K σ j ~P − z j +1 b ∗ ~k,λ = (A.50)= b ∗ ~k,λ ( ~ Γ σj~P + ~k ) + P λ R R | ~q | δ σ j ~P ( b q ) b ∗ ~q,λ b ~q,λ d q + E σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 . Moreover, for σ j +1 ≤ | ~k | ≤ σ j , j ≥
1, and for α , µ , and ǫ small enough but uniform in j , wecan control the following series expansion in the space F σ j ( ~ Γ σj~P ) + H fδ σj~P + E σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 × (A.51) × + ∞ X n =0 (cid:2) − ( ~ Γ σ j ~P · ~k + | k | ( ~ Γ σj~P ) + H fδ σj~P + E σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:3) n FP Infraparticle Scattering H fδ σj~P := X λ Z R | ~q | δ σ j ~P ( b q ) b ∗ ~q,λ b ~q,λ d q , the key estimate being (cid:13)(cid:13)(cid:13) (cid:16) ( ~ Γ σj~P ) + H fδ σj~P + E σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:17) / × (A.52) × ( ~ Γ σ j ~P · ~k + | k | (cid:16) ( ~ Γ σj~P ) + H fδ σj~P + E σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:17) / (cid:13)(cid:13)(cid:13) F σj ≤ c < . In order to control the term proportional to ~ Γ σ j ~P · ~k , we note that X i =1 (cid:13)(cid:13)(cid:13) (cid:16) K σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:17) / ( ~ Γ σ j ~P ) i √ (cid:13)(cid:13)(cid:13) F σj ≤ (cid:13)(cid:13)(cid:13) (cid:0) K σ j ~P + O ( α ) (cid:1) (cid:12)(cid:12)(cid:12) K σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) F σj . (A.53)Then, we observe thatlim sup µ,α,ǫ → (cid:16) | ~k | δ σ j ~P ( b k ) (cid:17) / (cid:13)(cid:13)(cid:13) (cid:0) K σ j ~P + O ( α ) (cid:1) (cid:12)(cid:12)(cid:12) K σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) / F σj √ | ~k |≤ sup ~P ∈S | ~P |√ − | ~P | ≤ √ . (A.54)Therefore, the estimate (A.52) also holds true for the term proportional to ~ Γ σ j ~P · ~k if µ > α >
0, and ǫ > j . To estimate of the term proportionalto | ~k | , we use | ~k | (cid:13)(cid:13)(cid:13) K σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:13)(cid:13)(cid:13) F σj ≤ | ~k | | ~k | δ σ j ~P ( b k ) − µσ j +1 ) ≪ , (A.55)for α , ǫ , µ small enough but unifor in j . Therefore, recalling that b ~k,λ Φ σ j ~P = 0 for | ~k | ≤ σ j , FP Infraparticle Scattering ≤ C n α X λ Z B σj \B σj +1 d k (cid:12)(cid:12)(cid:12) k l ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) + ( b l · ~ε ~k,λ ) q | ~k | ) (cid:12)(cid:12)(cid:12) × (A.57) × (cid:13)(cid:13)(cid:0) K σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:1) (Γ σ j ~P ) l Φ σ j ~P (cid:1)(cid:13)(cid:13) o + ∞ X n =0 c n ≤ − c C n α X λ Z B σj \B σj +1 d k (cid:12)(cid:12)(cid:12) k l ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) + ( b l · ~ε ~k,λ ) q | ~k | ) (cid:12)(cid:12)(cid:12) × (A.58) × (cid:13)(cid:13)(cid:0) K σ j ~P + | ~k | δ σ j ~P ( b k ) − z j +1 (cid:1) (Γ σ j ~P ) l Φ σ j ~P (cid:1)(cid:13)(cid:13) o ≤ − c C C (cid:0) α X λ Z B σj \B σj +1 d k (cid:12)(cid:12)(cid:12) k l ~ ∇ E σ j ~P · ~ε ~k,λ | ~k | δ σ j ~P ( b k ) + ( b l · ~ε ~k,λ ) q | ~k | ) (cid:12)(cid:12)(cid:12) (cid:1) × (A.59) × (cid:12)(cid:12) D (Γ σ j ~P ) l Φ σ j ~P , K σ j ~P − z j +1 (Γ σ j ~P ) l Φ σ j ~P E(cid:12)(cid:12) where, in passing from (A.58) to (A.59), we use (IV.67), and property ( A
3) from SectionIV D. For σ ≤ | ~k | ≤ σ , a similar argument yields (A.59).This proves the lemma. FP Infraparticle Scattering Lemma A.4.
For α and ǫ small enough, with ǫ > C α , C sufficiently large, there existconstants R , R ≤ O ( ǫ − ) , uniformly in j ∈ N and ~P ∈ S , for which (cid:13)(cid:13)(cid:13)(cid:16) K σ j − ~P − z j +1 (cid:17) (( b Γ σ j ~P ) i b Φ σ j ~P − (Γ σ j − ~P ) i b Φ σ j ~P ) (cid:13)(cid:13)(cid:13) (A.60)+ 8 (cid:13)(cid:13)(cid:13)(cid:16) K σ j − ~P − z j +1 (cid:17) (Γ σ j − ~P ) i ( b Φ σ j ~P − Φ σ j − ~P ) (cid:13)(cid:13)(cid:13) ≤ R ǫ j (cid:16) k b Φ σ j ~P − Φ σ j − ~P k + ǫ j ǫ j (cid:17) (A.61)+ R ǫ j (cid:16) (cid:13)(cid:13) b Φ σj~P k b Φ σj~P k − Φ σj − ~P k Φ σj − ~P k (cid:13)(cid:13) + ǫ j ǫ j (cid:17) . Proof.
In order to justify the estimate in the statement, it is enough to make the difference( b Γ σ j ~P ) i − (Γ σ j − ~P ) i (A.62)explicit. The definitions are given in (IV.41) and (IV.54).¿From (A.16), (A.18), we get − (cid:10) b Φ σ j ~P , b ~ Π σ j ~P b Φ σ j ~P (cid:11)(cid:10) b Φ σ j ~P , b Φ σ j ~P (cid:11) + (cid:10) Φ σ j − ~P , ~ Π σ j − ~P Φ σ j − ~P (cid:11)(cid:10) Φ σ j − ~P , Φ σ j − ~P (cid:11) (A.63)= ~ ∇ E σ j ~P − ~ ∇ E σ j − ~P (A.64)+ h W σ j ( ~ ∇ E σ j ~P ) ~β σ j W ∗ σ j ( ~ ∇ E σ j ~P ) i Ω f (A.65) −h W σ j − ( ~ ∇ E σ j − ~P ) ~β σ j − W ∗ σ j − ( ~ ∇ E σ j − ~P ) i Ω f . ¿From (A.26) – (A.31) and (A18) – (A25), we obtain b ~ Γ σ j ~P − ~ Γ σ j − ~P = b ~ Π σ j ~P − ~ Π σ j ~P (A.66) − (cid:10) b Φ σ j ~P , b ~ Π σ j ~P b Φ σ j ~P (cid:11)(cid:10) b Φ σ j ~P , b Φ σ j ~P (cid:11) + (cid:10) Φ σ j − ~P , ~ Π σ j − ~P Φ σ j − ~P (cid:11)(cid:10) Φ σ j − ~P , Φ σ j − ~P (cid:11) (A.67)= ~ ∇ E σ j ~P − ~ ∇ E σ j − ~P + ~ L σ j − σ j (A.68)+ α X λ Z B σj − \B σj ~k ~ ∇ E σ j − ~P · ~ε ∗ ~k,λ ~ ∇ E σ j − ~P · ~ε ~k,λ | ~k | ( δ σ j − ~P ( b k )) d k (A.69)+ α X λ Z B σj − \B σj (cid:2) ~ε ~k,λ ~ ∇ E σ j − ~P · ~ε ∗ ~k,λ | ~k | δ σ j − ~P ( b k ) + h.c. (cid:3) d k q | ~k | . (A.70) FP Infraparticle Scattering (cid:13)(cid:13)(cid:13) (cid:16) K σ j − ~P − z j +1 (cid:17) (Γ σ j − ~P ) i (cid:13)(cid:13)(cid:13) F σj ≤ O ( ǫ − j +12 ) (A.71) (cid:13)(cid:13)(cid:13) (cid:16) K σ j − ~P − z j +1 (cid:17) ~A σ j − σ j (cid:13)(cid:13)(cid:13) F σj ≤ O ( ǫ j − ) , (A.72)and similarly for ~ L σ j − σ j . The size of all other expressions (A.69) – (A.70) can trivially beseen to be of order O ( α ǫ j − ). The assertion of the lemma follows. Acknowledgements
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