Quasi-static Limits in Nonrelativistic Quantum Electrodynamics L. Tenuta 8 0 0 2 n Mathematisches Institut, Eberhard-Karls-Universit¨at, Auf der Morgenstelle a 10, 72076, Tu¨bingen, Germany. J e-mail: [email protected] 0 1 Abstract ] h We consider a system of N nonrelativistic particles of spin 1/2 p - interacting with the quantized Maxwell field (mass zero and spin one) h in the limit when the particles have a small velocity. t a Two ways to implement the limit are considered: c with the m → ∞ velocity v of the particles fixed, the case for which rigorous results [ have already been discussed in the literature, and v 0 with c fixed. 2 → The second case can be rephrased as the limit of heavy particles, v 5 mj ε−2mj, observed over a long time, t ε−1t, ε 0+, with → → → 1 kinetic energy E = (1). kin 2 O Focusing on the second approach we construct subspaces which 1 . are invariant for the dynamics up to terms of order ε log(ε−1) and 7 0 describeeffective dynamics, for the particles only, insidpethem. At the 7 lowestordertheparticlesinteractthroughCoulombpotentials. Atthe 0 second one, ε2, the mass gets a correction of electromagnetic origin : v and a velocity dependent interaction, the Darwin term, appears. i X Moreover, we calculate the radiated piece of the wave function, i. r e., the piece which leaks out of the almost invariant subspaces and a calculate the corresponding radiated energy. 1 Introduction A system of nonrelativistic particles of spin 1/2 interacting with the quan- tized radiation field is described by the so-called Pauli-Fierz Hamiltonian, or “nonrelativistic quantum electrodynamics”. The model is thought to have an extremely wide range of validity, apart from phenomena connected to 1 gravitational forces and from other ones typical of high-energy physics like pair creation, whose description requires the use of full relativistic QED. This belief is mainly based on the analysis of some formal limit cases, which can be accurately studied both from a theoretical and an experimen- tal point of view. Indeed, the interaction between charged particles is usually described by instantaneous pair potentials of Coulomb-type, without intro- ducing the field as dynamical variable. This is known to be a good approx- imation if the particles move sufficiently slowly. One aim of this paper is a mathematically rigorous justification of this fact, i. e., the derivation of the Schr¨odinger equation with Coulomb potentials, and the second order ve- locity dependent corrections to them, starting from nonrelativistic quantum electrodynamics. In addition, a formula is provided for the wave function of the radiated photons and the corresponding radiated energy, which is the quantum equivalent of the Larmor formula of classical electrodynamics. In more detail the model considered is given, excluding the addition of the electronic spin, by the canonical quantization of a system of N classical charges interacting through the Maxwell field. A sharp ultraviolet cutoff is introduced assuming that each charge has a charge distribution given by ̺ (x) = e ϕ(x), x R3, (1) j j ∈ where the form factor satisfies ϕˆ(k) = (2π)−3/2 for k Λ, 0 otherwise (note | | ≤ that there is no infrared cutoff). The classical equations of motion are given by 1 ∂ B(x,t) = E(x,t), c t −∇× N (2) 1 q˙ (t) j ∂ E(x,t) = B(x,t) e ϕ x q (t) , c t ∇× − j − j c Xj=1 (cid:0) (cid:1) with the constraints N E(x,t) = e ϕ x q (t) , B(x,t) = 0, (3) j j ∇· − ∇· Xj=1 (cid:0) (cid:1) and the Newton equations for the particles, q˙(t) l m q¨(t) = e [E (q (t),t))+ B (q (t),t)], l = 1,...N, (4) l l l ϕ l ϕ l c × where E (x,t) := (E ϕ)(x,t) and analogously for B . ϕ x ϕ ∗ 2 The canonical quantization of this system in the Coulomb gauge is de- scribed, e. g., in ([Sp], chapter 13). The Hilbert space of the pure states is given by H := H F. (5) p ⊗ The space for the particles, H , is defined by1 p H := L2(R3 Z )⊗N, (6) p 2 × where R3 is the configuration space of a single particle and Z represents its 2 spin. The state space for a single photon is L2(R3 Z ), where R3 is the mo- 2 × mentum space of the photon and Z represents its two independent physical 2 helicities. The photon Fock space is therefore F := ∞ M L2(R3 Z ), (7) ⊕M=0 ⊗(s) × 2 where M denotes the M-symmetric tensor product and 0 L2(R3 Z ) := ⊗(s) ⊗(s) × 2 C. We denote by Ω the vector (1,0,...), called the Fock vacuum. F The dynamics of the system are generated by the Hamiltonian N 2 1 1 Hc := σ i e A (x ) +V (x)+cH , (8) j j ϕ j ϕcoul f 2m (cid:20) ·(cid:18)− ∇− √c (cid:19)(cid:21) Xj=1 j where all the operators appearing are independent of c and we use units in which ~ = 1. σ is a vector whose components are the Pauli matrices of j the jth particle, A (x ) denotes the quantized transverse vector potential ϕ j in the Coulomb gauge, V is the smeared Coulomb potential and H the ϕcoul f free field Hamiltonian. The reader who is not familiar with the notation is advised to look at section 2, where the model is described in more detail. To implement practically the idea that the particles move “slowly”, a standard procedure, applied also in classical electrodynamics (see, e.g., [Ja], [LaLi]), is to take the limit c 2. Since c is a quantity with a dimension, → ∞ one should actually say that v /c 0, where v is a typical velocity of the | | → 1The formalism presented holds also in the case when all the particles are equal and their Hilbert space is given by the subspace of totally antisymmetric wave functions. In this case,the dipole radiationgivenin(29)iszero. We considerthereforethe generalcase of different particles. 2 Intheclassicalcase,amorerefinedandpreciseanalysisiscarriedthroughin[KuSp1], [KuSp2]. The authors consider, loosely speaking, initial conditions which represent free particles moving together with the field they generate (“dressed” particles or charge soli- tons), with a velocity of order (ε1/2) with respect to the speed of light. Assuming that the particles are at time t=0 Ofar apart (relative distance of order (ε−1)) and rescaling O 3 particles. This can be achieved in two ways, fixing v and letting c or → ∞ fixing c and letting v 0. → In the classical case this is reflected in the fact that the limit c is → ∞ equivalent, up to a rescaling of time, to the limit of heavy particles, as one can easily verify replacing in equations (2)-(4) m with ε−2m , t with ε−1t, l l and looking at the limit ε 0. → We will show that in the quantum case the two procedures are non equiv- alent anymore, a fact that can be intuitively explained by the presence of an additional scale given by ~. In this paper we concentrate on the limit of heavy particles observed over a long time. An additional aim is to point out similarities and differences between the two limits in the quantum context and to compare the results we get for the Pauli-Fierz model with the ones valid for the Nelson model, where the particles and the photons are spinless, [TeTe]. We recall briefly in the next subsection some known results about the case c and then illustrate in more detail the limit ε 0. → ∞ → 1.1 The limit c → ∞ This case, as observed by Spohn [Sp], has the form of a weak coupling limit, fact which was already noted for the Nelson model by Davies [Da ], who 2 formulated also a general scheme to analyze the limit dynamics in the weak coupling case[Da ](anextended notionofweakcoupling limitforPauli-Fierz 1 systems in which the Hilbert space of the particles is finite dimensional has been examined in [DeDe]). Davies looks loosely speaking at the limit λ 0 for the time evolution → generated by an Hamiltonian of the form λ−2(H +λH ), 0 int which corresponds physically to a weak interaction, whose effect is however observed over the long time scale λ−2. suitablythedynamicalvariables,theyshowthattheparticlesremainatarelativedistance of order (ε−1) for long times (of order (ε−3/2)) and on this time scale their motion is O O governed by effective dynamics. The possibility to implement an analogous limit in the quantumcaseisunclear,becausethereisnoobviousquantumcounterparttothe classical chargesolitons. ThePauli-FierzHamiltonianwithoutinfraredcutoffhasindeednoground state in the Fock space for fixed total momentum different from zero [Fr] [Ch]. We stick therefore to the more pragmatic choice c . The states which we define through th→e d∞ressing transformation U in equation (27) ε should be consideredapproximatedressedstates valid for smallvelocities of the particles. 4 The Hamiltonian Hc assumes a similar form if we consider it on the long time scale defined by c2. Putting λ := c−3/2 we get indeed c2Hc = λ−2(H +λ2/3H +λh +λ4/3h ) := λ−2H , (9) f p 1 4/3 λ where N 1 H := ∆ +V , (10) p j ϕcoul − 2m Xj=1 j N e e j j h := i A (x ) σ B (x ), (11) 1 m ∇xj · ϕ j − 2m j · ϕ j Xj=1 j j N e2 h := j : A (x )2 : , (12) 4/3 ϕ j 2m Xj=1 j where B := A and we normal order the quadratic term. ϕ ϕ ∇× Applying Davies scheme one gets in the end ([Sp], theorem 20.5) Theorem 1. Let ψ H1(R3N,C2N), then ∈ lim (e−iHcc2t e−iHdarwc2t)ψ ΩF H = 0, (13) c→∞k − ⊗ k where H := H +c−2V +c−2V , (14) darw p darw spin N e e ϕˆ(k) 2 V := j l dk | | eik·xji (1 κ κ)i e−ik·xl,(15) darw − m m Z 2 k 2 ∇xj · − ⊗ ∇xl jX,l=1 j l R3 | | N e e V := j l σ σ dk ϕˆ(k) 2eik·(xj−xl), (16) spin j l −X 12mlmj · ZR3 | | j,l=1 where (κ κ) := κ κ and κ := k/ k . ij i j ⊗ | | V gives rise to a correction of electromagnetic origin to the mass of darw the particles and to a velocity dependent potential, called the Darwin term. It already appears in classical electrodynamics when the dynamics of the particles are expanded up to terms of order (v/c)2 (see, e. g., [Ja] or [LaLi]). 5 For the convenience of the reader and to ease the comparison with the results for the limit of heavy masses we give a formal derivation of this theorem in appendix A. We note here that the method employed in the weak coupling case forces one to consider as initial condition for the field just the Fock vacuum, which contains no photons at all. There is therefore no analogy with the physical picture that every particle should be described by a “dressed state”, loosely speaking the particle itself dragging with it a cloud of “virtual” photons. 1.2 The limit m → ∞ The situation is different in the case m , which is more conveniently j → ∞ studied adopting units where c = 1. Replacing m by ε−2m we get then the j j Hamiltonian N 1 e e Hε := pˆ2 +V +H ε j pˆ A (x ) ε2 j σ B (x )+ 2m j ϕcoul f − m j · ϕ j − 2m j · ϕ j Xj=1 j j j e2 +ε2 j : A (x )2 : , ϕ j 2m j (17) where we have indicated with pˆ the ε-momentum of the jth particle j pˆ := iε . (18) j − ∇xj As already pointed out talking about the classical case, the dynamics have to be observed over times of order (ε−1). This is necessary in order to O see non trivial effects, because we consider initial states with bounded kinetic energy. Since the particles have a mass of order (ε−2) this means that their O velocity is in the original time scale of order (ε). O Toanalyzethelimitε 0weconstruct aunitarydressing transformation → U : H H , which allows us to define dressed states for small velocities ε → of the particles and to introduce a clear notion of real and virtual photons. More precisely, in the new representation defined by U , the vacuum sector ε H Ω corresponds to states of dressed particles without real photons, p F ⊗ while in the original Hilbert space a state with M real photons is a linear combination of states of the form U −1(ψ a(f )∗ a(f )∗Ω ), where ψ H , f ,...,f L2(R3 Z ). ε ⊗ 1 ··· M F ∈ p 1 M ∈ × 2 The projector on the subspace corresponding to dressed particles with M real photons is therefore Pε := U ∗(1 Q )U , M ε p ⊗ M ε 6 where Q denotes the projector on the M-particles subspace of the Fock M space. In short, we will show that the subspaces Pε H are approximately in- M variant for the dynamics defined by Hε on times of order (ε−1). Moreover, O on this time scale we will give effective dynamics for states inside such a sub- space, with an error of order (ε2log(ε−1)). The effective dynamics contain O the Darwin correction described in (15), but no spin dependent term. One can get an idea of why this happens comparing the expression of Hε with that ofH , equation(17) and(9). InHε thespin dependent termis ofsecond λ order, while in H is of the first one. In the limit ε 0 the analogue of V λ spin → would be of order (ε4), therefore it does not appear in an expansion of the O time evolution till second order. Finally we compute the leading order part of the state which makes a transition between Pε and Pε, which corresponds 0 1 to the emission of one real photon. The corresponding radiated energy is given by a quantum analogue of the Larmor formula. The procedure to construct the unitary U is explained in detail in [TeTe] ε for the Nelson model. The technique used is based onspace-adiabatic pertur- bation theory [Te], a method which allows to expand the dynamics generated by a pseudodifferential operator with an ε-dependent semiclassical symbol. The main difficulty in all models concerning the interaction of particles with a quantized field of zero mass is that, because of soft photons, the principal symbol of the Hamiltonian has no spectral gap, which is a condition required to apply the methods of [Te]. In the case of Hε we have indeed N 1 h (p,q) := p2 +V (q)+H , (p,q) R3N R3N . (19) 0 2m j ϕcoul f ∈ × Xj=1 j For every fixed (p,q) this is an operator on F and has a ground state given by Ω , at the threshold of the continuous spectrum. The corresponding F eigenvalue N 1 E (p,q) = p2 +V (q) (20) 0 2m j ϕcoul Xj=1 j isthesymbolofanHamiltonianactingjustonH anddescribingtheparticles p interacting through the smeared Coulomb potential. The trouble connected to the absence of the spectral gap is solved by introducing an effective gap, considering the Hamiltonian Hε,σ where the form factor ϕˆ (see equation (1)) is replaced by ϕˆ (k) := (2π)−3/2 for σ < σ k < Λ, 0 otherwise. | | Proposition. (see proposition 3) 7 Suppose that the cutoff σ is a function of ε, σ = σ(ε), such that σ(ε) < ε2, then ke−itHε/ε −e−itHε,σ/εkL(H0,H) ≤ C|t|σ(ε)1/2 (21) where H := D Hε) (22) 0 0 (cid:0) is the domain of the free Hamiltonian N 1 Hε := pˆ2 +H (23) 0 2m j f Xj=1 j with the corresponding graph norm. Fixing σ, e. g., as a sufficiently high power of ε we can then replace the original dynamics with infrared cutoff ones. For Hε,σ it is possible to build a dressing operator U which can be ε,σ expanded in a series of powers of ε with σ-dependent coefficients which are atmostlogarithmicallydivergent. UsingitwedefinethedressedHamiltonian Hε,σ := U Hε,σU ∗ (24) dres ε,σ ε,σ which can be expanded in a series of powers of ε in (H ,H ), with coeffi- 0 L cients which are also at most logarithmically divergent in σ. The different coefficients in the expansion correspond to different physical effects which can be now clearly separated according to their order of magnitude in ε. Thefirstresultwefind, aswealreadymentionedabove, isthatthedressed M-photons subspaces are approximately invariant for the dynamics: Theorem. (see corollary 2). Given a χ C∞(R) and a function σ(ε) such that ∈ 0 ε−2σ(ε)1/2 0, ε log(σ(ε)−1) 0, ε 0+, (25) → → → p then k[e−iHεεt,PMε ]χ(Hε)kL(H) = O √M +1|t|ε log(σ(ε)−1) , (26) (cid:0) p (cid:1) where Pε := U ∗ (1 Q )U . (27) M ε,σ(ε) p ⊗ M ε,σ(ε) The adiabatic decoupling which guarantees the invariance of the sub- spaces holds uniformly only on states in which the particles have a uniformly bounded kinetic energy. For this reason we introduce a cutoff function on the 8 total energy χ, which gives rise automatically to a bounded kinetic energy for the slow particles. In the following we assume that the function σ(ε) has been fixed so that (25) is satisfied. One can then approximate the dynamics of the particles inside each almost invariant subspace. Theorem. (see theorem 6). Let S be a bounded observable for the particles, S (H ), and ω p ∈ L ∈ I (Pε χ(Hε)H ) a density matrix for a mixed dressed state with M free 1 M photons, whose time evolution is defined by ω(t) := e−itHε/εωeitHε/ε . We have then TrH S 1F ω(t) = TrH Se−itHeεffTrF(ω)eitHeεff + (cid:18) ⊗ (cid:19) p(cid:18) (cid:19) (cid:0) (cid:1) + (ε3/2 t )(1 δ )+ ε2log(σ(ε)−1)( t + t 2) , M0 O | | − O | | | | (cid:0) (cid:1) where δ = 1, when M = 0, 0 otherwise, and M0 N 1 Hε := pˆ2 +V + eff 2m j ϕcoul Xj=1 j N e e ϕˆ(k) 2 ε2 j l dk | | eik·xjpˆ (1 κ κ)pˆe−ik·xl = (28) − m m Z 2 k 2 j · − ⊗ l lX,j=1 j l R3 | | N 1 = pˆ2 +V +ε2V . 2m j ϕcoul darw Xj=1 j Remark 1. Even though the subspaces Pε depend on the choice of the in- M frared cutoff, the effective Hamiltonian is infrared regular and therefore in- dependent of σ. Moreover, as we briefly mentioned above, it contains the corrections to the mass of the particles and the Darwin term, but no spin dependent term (compare with theorem 1). This topic is further discussed in the proof of theorem 5 and in remark 4. Since the subspaces Pε are only approximately invariant, there is a piece M of the wave function which “leaks out” in the orthogonal complement. This correspond physically to the emission or absorption of free photons. For a system starting in the dressed vacuum the leading order of the wave function of the emitted photon is given in the next theorem. 9 Theorem. (see corollary 3). Up to terms of order ε2log(σ(ε)−1)( t + t 2) , the radiated piece for a O | | | | system starting in the dress(cid:0)ed vacuum (M = 0) is g(cid:1)iven by Ψrad(t) := (1−P0ε)e−iεtHεP0εχ(Hε)Ψ ∼= iε ϕˆ (k) t (29) ∼= −e−ithˆε0√2 σk(ε3)/2 eλ(k)·Z dsei(s−t)|k|/εOpWε D¨(s;x,p) ψ(x), | | 0 (cid:0) (cid:1) where e (k) is the polarization vector of a photon with helicity λ, λ N 1 hˆε := pˆ2 +V +H , (30) 0 2m j ϕcoul f Xj=1 j ψ(x) :=< ΩF,χ(hˆε0)Ψ >F ∈ Hp, (31) N e D(s;x,p) := j xcl(s;x,p), (32) m j Xj=1 j OpW denotes the Weyl quantization acting on a suitable symbol space on ε R3N R3N and xcl is the solution to the classical equations of motion × j m x¨cl(s;x,p) = V (xcl(s;x,p)), j j −∇xj ϕcoul xcl(0;x,p) = x , x˙cl(0;x,p) = p m−1, j = 1,...,N . j j j j j Remark 2. As explained in detail in remark 5, generically the norm of the radiated piece is bounded below by εlog(εσ(ε)−1) , which means that O the subspace Pε is near optimal, i. e. t(cid:0)he transitions(cid:1)are at least of order 0 εlog(εσ(ε)−1) . O (cid:0)Note that, lik(cid:1)e in classical electrodynamics, when all the particles are equal, the leading order of the radiated piece vanishes, because D is then proportional to the position of the center of mass, whose acceleration is zero. Remark 3. Even though the radiated wave function has no limit when ε 0, → because ϕ(k) k −3/2 / L2(R3), the corresponding radiated energy has a limit. | | ∈ Defining E (t) := Ψ (t),H Ψ (t) , (33) rad rad f rad h i we get to the leading order (see remark 6) d ε3 Prad(t) := dtErad(t) ∼= 3π2hψ,OpWε |D¨(t)|2 ψiHp . (34) (cid:0) (cid:1) 10