Electromagnetic gauge choice for scattering of Schr\"odinger particle

We consider a Schr\"{o}dinger particle placed in an external electromagnetic field of the form typical for scattering settings in the field theory: $F=F^\mathrm{ret}+F^\mathrm{in}=F^\mathrm{adv}+F^\mathrm{out}$, where the current producing $F^{\mathrm{ret}/\mathrm{adv}}$ has the past and future asymptotes homogeneous of degree $-3$, and the free fields $F^{\mathrm{in}/\mathrm{out}}$ are radiation fields produced by currents with similar asymptotic behavior. We show that with appropriate choice of electromagnetic gauge the particle has 'in' and 'out' states reached with no further modification of the asymptotic dynamics. We use a special quantum-mechanical evolution 'picture' in which the free evolution operator has well-defined limits for $t\to\pm\infty$, thus the scattering wave operators do not need the free evolution counteraction. The existence of wave operators in this setting is established, but the proof of asymptotic completeness is not complete: more precise characterization of the asymptotic behavior of the particle for $|\mathbf{x}|=|t|$ would be needed.


Introduction
Infrared problems are typical for theories with long-range interactions, and extend over wide range of physical settings. They are particularly persistent in the relativistic quantum theory-quantum field theory-where their nature is not only technical, but also conceptual. The standard procedure adopted in mathematically oriented formulations of the quantum electrodynamics (and other theories with long range interaction) is to use local potentials (of Gupta-Bleuler type) with adiabatically truncated interaction. One argues that this setting is sufficient to construct (perturbatively) the local algebra of observables of the theory. However, the removal of the cut-off is a singular operation, which has the consequence that the states of the desired theory cannot be those in which the truncated theory is constructed. This is well-known and generally accepted. But one can also ask, whether the truncation of interaction does not remove some structure from the theory in an irreversible way; the algebraic equivalence of local algebras, for all cutoff functions equal to one in the considered region, is based on an interpolating relation, which becomes singular in the limit of the function tending to unity on the whole spacetime. 1 Considerations similar to those described above has motivated the present author, many years ago, to attempts to include, from the beginning, the long-range degrees of freedom of the theory in the description. These attempts went in two main directions: (i) construction of a nonlocal electromagnetic potential in which scattering of a Dirac particle is infrared nonsingular (see [4] for the analysis at the level of classical fields); and (ii) extension of the algebra of the free quantum electromagnetic field which includes long-range degrees of freedom, and which is thought of as an asymptotic algebra, potentially starting point for perturbation calculus (see [5] and a recent synopsis [6]).
The present article is a further test of the idea mentioned in (i) above. In article [4] I have considered the scattering of the classical Dirac field in an external electromagnetic field of the type supposed to be present in full interacting theory. It was shown that if an appropriate (nonlocal, in general) electromagnetic gauge is chosen, then the Dirac field has a well-defined asymptote in remote future (and past) inside the lightcone. This asymptote is reached without further corrections of asymptotic dynamics, as usually employed in long-range scattering. Moreover, the scattered outgoing Dirac field is constructed with the use of this asymptote. The hope behind this analysis is that a similar construction in quantum case could similarly relieve some of the infrared problems.
The article mentioned above lacks the discussion of the asymptotic behavior of the Dirac field on the whole hyperplanes of constant time, for this time tending to infinity. In the present paper we analyze this question in the case of nonrelativistic Schrödinger particle. We show the existence of asymptotic velocity operators and of isometric wave operators. The null asymptotic behavior of radiation fields (and their potentials), together with all their derivatives, is only of (1/time) type, which breaks the usual assumptions imposed on time-decaying potentials considered in Schrödinger scattering (see [2], [11]). In the present setting this behavior has prevented the proof of asymptotic completeness. Whether this can be overcome is an open question. 2 The choice of gauge found appropriate for the problem of [4] was such that x¨Apxq (Minkowski product of the position vector with the potential) vanishes sufficiently fast in remote future (and past) inside the lightcone (x 2 ą 0). Here we shall construct an appropriate gauge in the whole spacetime, and will see that its behavior inside the lightcone is of the same type as before.
The method used for our analysis is the transformation of the time evolution from the Schrödinger picture to a new 'picture', which we describe in Section 2, and in which the state vector of a particle tends to its spacetime asymptotic forms in asymptotic times. As it turns out, the simplest choice of such transformation is the well-known Niederer transformation to the harmonic oscillator system [8]. In Section 3 this oscillator is placed in electromagnetic field, and in Section 4 transformed back to the Schrödinger picture. In order to construct the time-dependent hamiltonian, and the evolution it generates, we follow an article by Yajima [12] in its use of the Kato theorem. 3 In Section 5 we discuss in detail the relation between potentials and fields in the two pictures, and define a gauge appropriate for the description of scattering along the lines described above. Section 6 contains theorems on scattering in oscillator picture. Reformulation of these results in natural spacetime terms and some final remarks are contained in Section 7. Appendix A discusses some relations between domains of operators needed in the main text. In Appendix B we discuss the scope of electromagnetic fields admitted in the article, and analyse, for completeness, their spacetime estimates. Appendix C clarifies a particular differentiation employed in Section 6.

Harmonic oscillator picture
Let H " L 2 pR 3 q and denote H 0 "´1 2 ∆, the free particle hamiltonian with the corresponding evolution operator U 0 pt, sq. 4 The free particle wave packet ψptq " U 0 pt, 0qψ is given by the Fourier representation and forψ regular enough, by the use of stationary phase method, has the following asymptotic forms for |t| Ñ 8: We would like to find a new evolution 'picture', in which the state vector of the particle has well defined limits for t Ñ˘8. Using the asymptotic forms 3 There is also some similarity in the use of two time evolutions related by unitary transformations. However, the transformations are quite different: it is a pure gauge transformation in [12], while here it is the transformation to the new 'picture' mentioned above. Also, the aims of these operations are quite different: Yajima's goal is to include electromagnetic fields as singular as possible (both locally, and in infinity), while here we are interested in scattering theory. 4 We set " 1, c " 1 and use dimensionless rescaled quantities; to recover physical quantities one should substitute pt, xq Þ Ñ pmt, mxq, with m the mass of the particle. Also, the electromagnetic potentials to appear later should be multiplied by q{m, with q the charge of the particle.
(1) as a guideline, we define for t P R the following transformation in H: where xty ą 0 is a C 1 -function of t, such that xty{|t| Ñ 1 for |t| Ñ 8, whose exact form is to be determined. This is a unitary transformation and the conjugate transformation is rN ptq˚ψspyq " xty 3{2 exp "´i 2 t|y| 2 ‰ ψpxtyyq .
We use the family of operators N ptq to transform the Schrödinger state vectors and observables respectively by The evolution operator for ψ N ptq: is strongly continuous, and for χ P C 8 0 pR 3 q the vector U 0N pt, sqχ is strongly differentiable in t and in s, and a straightforward calculation shows that 1 2xty 2´r p 2`p xty 2´t2 qx 2 s`pt´xtyxty 1 qrx¨p`p¨x`2tx 2 s¯; here and in what follows p "´iB B B. A large class of functions xty for which t´xtyxty 1 tends to zero and xty 2´t2 tends to a constant sufficiently fast would fulfil our demands formulated after (1). However, the simplest choice is to demand t´xtyxty 1 " 0, in which case xty 2 " t 2`κ2 , where κ 2 is a constant. The choice of this constant is physically irrelevant, and as our variables are dimensionless, the simplest choice is κ " 1, so that N p0q " 1.
The new picture hamiltonian is now and the choice of xty holds in the rest of the article. If we now define u 0 pτ, σq " U 0N ptan τ, tan σq , The evolution of the free particle state over t P p´8,`8q corresponds now to the evolution over τ P p´π{2,`π{2q, thus over half the period of the harmonic oscillator. We shall call this new description of evolution the harmonic oscillator picture. The scattering change of state is described in this picture by the operator where P is the parity operator rP χspxq " χp´xq. This formula is in agreement with the asymptotic forms (1). The transformation N ptq leading from a free particle to an harmonic oscillator has been discovered much earlier by Niederer [8]. It is interesting to note, that his original derivation was a result of group-theoretical considerations, with no relation to scattering theory. Although the Niederer transformation has found numerous applications in literature, to our knowledge it has not appeared in the scattering of a Schrödinger particle context before. Which, if true, would be quite surprising, if one notes its striking similarity to the asymptotic form (1).
We shall later see that the above relation extends to a system placed in external electromagnetic field: a charged particle dynamics becomes a charged oscillator in the new picture (with electromagnetic potentials appropriately transformed). For technical reasons, we find it convenient to start our discussion in the oscillator picture, and only then transform into Schrödinger picture.

Harmonic oscillator in electromagnetic field
We consider in H " L 2 pR 3 q the hamiltonian h " 1 2 pπ 2`x2 q`v , π " p´a , where apτ, xq and vpτ, xq are electromagnetic potentials. This hamiltonian may be given the precise meaning for a wide class of potentials (see [12]), but for our purposes it is sufficient to consider the following setting.
We also note an elementary inequality which will be used below. Let ψpτ q P H depend differentiably on a real parameter τ . Then To show this, note that B τ }ψpτ q} 2 " 2 Repψpτ q, B τ ψpτ qq and use the Schwarz inequality for the rhs.
If, in addition, pB B B¨aq is also in L 8 pR 3 q, then Dphq " Dph 0 q, and C 8 0 pR 3 q is a domain of essential self-adjointness of h.
Proof. For ϕ P Dpq 0 q one has |ρpϕ, ϕq| ď }aϕ}}pϕ}`pϕ, r 1 2 a 2`| v|sϕq ď 1 2 q 0 pϕ, ϕq`c 1 }ϕ} 2 . Therefore, the first part of the theorem follows by the KLMN theorem (see e.g. [10], Thm. X.17). If, in addition, pB B B¨aq P L 8 pR 3 q, then for ϕ P Dpq 0 q and ψ P Dph 0 q we have ρpϕ, ψq " pϕ, rψq with Thus the second part of the thesis follows by the Kato-Rellich theorem.T he existence of the corresponding evolution operators is assured under the conditions of the following theorem. Here and in the rest of the article the overdot denotes differentiation with respect to time in the oscillator picture.
Then (8) says that hpτ q : Y Þ Ñ X is norm-continuous. Moreover, let Y τ be Dph 0 q equipped with the graph norm }.} τ , and identify Apτ q " hpτ q. Then all conditions of this theorem are satisfied and one obtains upτ, σq with the stated properties.4 . From oscillator to Schrödinger picture The way back to the Schrödinger picture is achieved as follows.
Proof. Taking into account relations (2) one finds N psq˚Dph 0 q " Dph 0 q and N ptqDph 0 q " Dph 0 q, which proves (i). For ψ P Dph 0 q and F -the operator of multiplication by the function F pxq, we have so a straightforward calculation gives rN ptqhparctan tqN ptq˚ψspxq with A and V given in the thesis. On the other hand irN ptqB t N ptq˚ψspxq " 1 2 xty´2r´x 2`t px¨p`p¨xqsψpxq . Thus equations (11) are satisfied with Hptq " xty´2N ptqhparctan tqN ptq´iN ptqB t N ptq " 1 2 Πptq 2`V ptq . For each t the norm }Apt, .q} 8 is finite, and if }V pt, .q} 8 is finite as well, then the essential self-adjointness follows easily in standard way (as in the proof of Proposition 1). In general, there is only }x|x|y´1V pt, .q} 8 ă 8 (due to the x¨a term in V ), but then the use of the Leinfelder-Simader theorem (Thm. 4 in [7]) leads to the same conclusion.5

. Electromagnetic fields and gauges
We shall now discuss in detail the relation between pV, Aq and pv, aq defined in (9). We assume that all differentiations to appear may be performed, but in the following theorem the assumptions of Theorem 3 are not needed. Proposition 4. Let pV, Aq and pv, aq be related by (9). Denote the electric and magnetic fields of these potentials by pE, Bq and pe, bq, respectively. Then the following is satisfied: which we shall call an a-gauge. In these gauges the assumptions of Theorem 3 are reduced to the following: a e p0q, B B B¨a e p0q P L 8 pR 3 q and e, B B B¨e P L 8 pIˆR 3 q for each compact interval I Ă p´π{2,`π{2q. (iv) In all a-gauges the four-potential A a pxq " pV pt, xq, Apt, xqq satisfieŝ (there is no singularity at x 0 " 0).
Proof. All properties follow by simple calculations, which we leave to the reader.P reparing to discuss scattering, we need to formulate the asymptotic behavior of potentials and fields: for upper case fields in the limit |t| Ñ 8, and the corresponding behavior of lower case fields in the limit |τ | Ñ π{2. It will be convenient to denote Cpt, xq " tEpt, xq`xˆBpt, xq , In what follows we shall always use a-gauges, so we only need to consider the fields E, C and e. In Appendix B we summarize the asymptotic properties of electromagnetic fields in scattering settings in Minkowski space. For E and C it will be sufficient to note |Ept, xq| ď const , |Cpt, xq| ď const 1`|t|`|x| , while for divergence of C we shall need the bound for some 1 ě γ ą 0. A straightforward calculation with the use of (16) shows that these relations imply the following lower case fields bounds: |epτ, xq| ď const cos τ , |B B B¨epτ, xq| ď const pcos τ q 2´γ pcos τ`|| sin τ |´|x||q γ , (19) where in the last formula we have taken into account that B B B¨Ept, xq " 4πρpt, xq (the charge density), which gives a contribution to B B B¨e bounded by a constant, which is less restricting than the term in (19).
While reading the next section, the reader is asked to note that the bounds (19) are sufficient to satisfy the assumptions of Theorems 5 and 6, but for Theorem 7 the second bound in (19) would have to be replaced by |B B B¨epτ, xq| ď const pcos τ q 2´γ . (20)

Scattering in oscillator picture
Turning to the discussion of scattering, we start again with the oscillator picture. With increasingly restrictive assumptions we shall prove: (i) existence of asymptotic position, (ii) existence of wave operators, and (iii) their unitarity (asymptotic completeness). From now on we shall use only a-gauges defined by (13), and we omit the subscript e. For any operator k " kpτ q with Dph 0 q Ď Dpkpτ qq we denotẽ k "kpτ q " up0, τ qkpτ qupτ, 0q , so Dph 0 q Ď Dpkpτ qq .
We assume that the assumptions of Theorem 3, as reformulated in Proposition 4 (iii), are satisfied. Then for ψ P Dph 0 q, following the remarks in Appendix C, one finds 9 xψ "πψ , Noting that the integration by parts gives ż π{2 we observe that the rhs of (24) and (26) are integrable over r0, π{2s. Therefore, the limit lim τ Ñπ{2 expriq¨xpτ qsψ exists for each ψ P Dph 0 q, hence for all ψ. The existence of a self-adjoint operatorx`satisfying the limiting relation (25) follows. 7 The case ofx´is similar.W e now turn to the existence of wave operators.
The proof of the remaining claims is based on the following property, to be proved below: for each smooth function f pxq with compact support K not intersecting x 2 " 1, there is Once this is proved, we have where the first equality is the result of Theorem 5. Each bounded continuous function with support outside x 2 " 1, as well as the characteristic function of the set R 3 ztx|x 2 " 1u, is a point-wise limit of the above functions, so (30) and (31) follow (for the latter note that 1 t1u px 2 q " 0). Finally, there is }upτ, 0qE˘ψ´ω˚ψ} 2 " 2}E˘ψ} 2´2 RepE˘ψ, up0, τ qω˚ψq Ñ 0 for τ Ñ˘π{2, so (32) follows.
Turning to the proof of the remaining property (34) we use (6) to obtain for ψ P Dph 0 q: where u 0 and h 0 are unperturbed operators, and without restricting generality we assume }f } 8 ď 1. Now, f h´h 0 f " piB B Bf´f aq¨π`1 2 f piB B B¨a´a 2 q`ipB B Bf¨aq`1 2 p∆f q .
Electromagnetic gauge choice for scattering of Schrödinger particle 13 The norm of the integrand in (35) may be thus bounded as }rf hpσq´h 0 f supσ, 0qψ} ď p}B B Bf } 8`} apσ, .q} 8 q}πpσqψ} The integrability on r0, π{2s of the terms in the second line follows from the proof of the existence of ω`, and of the first term in the first line-from the proof of Thm. 5. Finally, the proof of integrability of }apσ, .q} 8 }πpσqψ} is very similar to the case of }apσ, .q} 2 8 (see relations (13) and (24)). This shows that the lhs of (35) has a limit for τ Ñ π{2. But u 0 p0, τ q has a unitary strong limit operator u 0 p0, π{2q, so f pxqupτ, 0qψ converges strongly to some vector χ. It follows that for each vector ϕ there is pup0, τ qf pxq˚ϕ, ψq " pω`f pxq˚ϕ, ψq , so χ " f pxqω˚ψ and the property (34) follows.I t should be clear from the proof of the above theorem that the only potential obstacle to the asymptotic completeness is the behavior of B B B¨epτ, xq in the neighborhood of x 2 " 1. If the norm }B B B¨epτ, .q} K,8 may be replaced by }B B B¨epτ, .q} 8 , then in the proof of convergence of (35) the function f may be replaced by 1, and one obtains the following.

Back to spacetime and conclusions
Let us remind the reader, that the class of scattering electromagnetic fields F pxq was announced in the abstract, and then discussed in Appendix B. The bounds they satisfy were adapted to the field e in Section 5, where we also anticipated that the resulting estimates satisfy the assumptions of Theorems 5 and 6. We now formulate the results of these theorems in natural spacetime terms, with the use of relation (10) providing the link between the oscillator picture evolution operator upτ, σq and the Schrödinger operator U pt, sq. The asymptotic variablesx˘provided by Theorem 5 may be obtained by where we used relation (12). Now they have a natural interpretation of asymptotic velocity operators. The wave operators ω˘of Theorem 6 are now ω˘" s-lim tÑ˘8 U p0, tqN ptq , and their conjugates are given by ω˚" s-lim tÑ˘8 N ptq˚U pt, 0qE˘.
It is visible that with our choice of gauge the definition of wave operators in the present formalism does not need further corrections (e.g. of the Dollard type) to compensate the long-range character of the interaction. The joint spectrum ofx`(and similarlyx´) covers the whole space R 3 and outsidẽ x 2 " 1 (x 2 " 1) is absolutely continuous. Whether this continuity extends to R 3 , in which case E˘" 1 and the asymptotic completeness holds, is an open problem. The difficulty lies in the null-asymptotic behavior of electromagnetic potentials (and fields), which is slower than assumed in usual analyses of timedecaying potentials. The problem may be due to the usual inconsistency in systems of the considered type: the Lorentz symmetry of the electromagnetic field vs Galilean symmetry of the Schrödinger equation.
The existence of the wave operators in the given form, despite the problems with asymptotic completeness, is a further argument for our main point: appropriate choice of gauge eliminates at least some of the infrared problems. The gauge condition found suitable in the present context is characterized by Eq. (14). This property closely parallels the gauge condition obtained in the analysis of the asymptotic behavior of the Dirac field inside the lightcone [4]. In our opinion this may have interesting implications for quantum electrodynamics as well, although in that context the non-locality of the gauge will be a problem (cf. (13)).

Acknowledgements
I am grateful to Jan Dereziński for important literature hints and to Paweł Duch for an interesting discussion.

ppendix B. Asymptotic behavior of electromagnetic fields
We discuss here the decay properties of electromagnetic fields in the Minkowski spacetime language. For more extensive discussion and explanation of details we refer the reader to [6].
Fields E and B are the 3-vector parts of the Minkowski tensor F ab pxq given by here and in the rest of these remarks: x " pt, xq; a, b are Minkowski indices and i, j, k are 3-space indices. Also, we note that the field C defined by (15) is the 3-vector space part of the field Electromagnetic fields present in scattering situations of the field theory have the form F " F ret`F in " F adv`F out , where F ret {F adv is the retarded/advanced field of an electromagnetic current: and F in {F out is a free incoming/outgoing field. In remote past and future the current J is assumed to tend to asymptotic currents, homogeneous of degree´3, with support inside the lightcone. Similarly, the free fields are produced as radiation fields of currents with the asymptotic behavior of the same type. We discuss the asymptotic behavior of fields in the half-space t ě 0 with the use of representation F " F adv`F out ; the case t ď 0 with F " F ret`F in is analogous. For t ě 0 and t`|x| tending to infinity, the dominant contribution to the advanced field comes from the asymptotic outgoing current. Therefore, in this region this field is homogeneous of degree´2, hence the field C adv is homogeneous of degree´1, and the field B B B¨C adv is homogeneous of degreé 2. Thus for the advanced part the bounds (17) and (18) are satisfied. In fact, the´2 homogeneity of B B B¨C adv implies that for the corresponding part e adv of e there is |B B B¨e adv pτ, xq| ď const{ cos τ , which is a bound of the type (20) sufficient for the validity of Thm. 7. 8 For the free outgoing field and its Lorenz gauge potential we use the integral representations valid for solutions of the wave equation (see e.g. [6]): A out a pxq "´1 2π ż 9 V a px¨l, lq d 2 l .
Here l represents a vector on the future lightcone and d 2 l is the Lorentzinvariant measure on the set of future null directions, applicable to functions of l homogeneous of degree´2. The function V describes the future null asymptote of the potential and the field: lim RÑ8 RA out px`Rlq " V px¨l, lq , lim RÑ8 RF out px`Rlq " 2l^9 V px¨l, lq , and 9 V ps, lq " BV ps, lq{Bs. Moreover, V ps, lq is homogeneous of degree´1, l¨V ps, lq " 0, V ps, lq Ñ 0 for s Ñ 8, and 9 V ps, lq (together with its low orders derivatives in the cone variable) is bounded by constp1`|s|q´1´ε for some ε ą 0. Using this bound it is easy to show that |A out pxq| ď const ż dΩplq p1`|x 0´x¨l |q 1`ε ď const 1`|x 0 |`|x| " θpx 2 q p1`|x 0 |´|x|q ε`θ p´x 2 q ı , where l represents unit 3-vectors and dΩplq " sin ϑdϑdϕ is the solid angle measure, with ϑ measured with respect to x. In the region x 2 ě 0 the field F out is estimated similarly as above (with the bound of : V ps, lq by constp1`|s|q´2´ε following from earlier assumptions), but in the region x 2 ă 0 one first has to integrate in (40) once by parts with respect to ϑ angle variable. In this way one finds |F out pxq| ď const 1`|x 0 |`|x| " 1 p1`||x 0 |´|x||q 1`ε`θ p´x 2 q 1`|x 0 |`|x| ı .
The estimate of E out in (17) is obviously satisfied. We turn to the field C out .
The relation (39) shows that this is again a solution of the wave equation with the integral representation C out a pxq "´1 2π ż 9 W a px¨l, lq d 2 l .
Denote L ab " l a B{Bl b´l b B{Bl a -the intrinsic differential operator on the lightcone. With the use of identity ş L ab F plqd 2 l " 0 valid for each C 1 -function F , homogeneous of degree´2, one shows that W a ps, lq " V a ps, lq´s 9 V a ps, lq´L ab V b ps, lq .
Now, it is easy to see that the function 9 W ps, lq has the same decay properties in |s| as function 9 V ps, lq, so C out pxq and B B B¨C out pxq are similarly estimated as A out pxq and F out pxq, respectively. This is sufficient to satisfy the bounds (17) and (18).