# Electromagnetic Gauge Choice for Scattering of Schrödinger Particle

## Abstract

We consider a Schrödinger 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 \(t\rightarrow \pm \infty \), and 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.

## Mathematics Subject Classification

Primary 81U99 Secondary 81V10## 1 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 cutoff 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 have motivated the present author, many years ago, to attempt 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 \(x\cdot A(x)\) (Minkowski product of the position vector with the potential) vanishes sufficiently fast in remote future (and past) inside the lightcone (\(x^2>0\)). 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 Sect. 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 Sect. 3 this oscillator is placed in electromagnetic field, and in Sect. 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 Sect. 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 given in Sect. 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 analyze, for completeness, their spacetime estimates. Appendix C clarifies a particular differentiation employed in Sect. 6.

## 2 Harmonic Oscillator Picture

^{4}The free particle wave packet \(\psi (t)=U_0(t,0)\psi \) is given by the Fourier representation

*t*, such that \(\langle t\rangle /|t|\rightarrow 1\) for \(|t|\rightarrow \infty \), whose exact form is to be determined. This is a unitary transformation and the conjugate transformation is

*N*(

*t*) to transform the Schrödinger state vectors and observables, respectively, by

*t*and in

*s*, and a straightforward calculation shows that

*the harmonic oscillator picture*. The scattering change of state is described in this picture by the operator

*P*is the parity operator \([P\chi ](\mathbf {x})=\chi (-\mathbf {x})\). This formula is in agreement with the asymptotic forms (1).

The transformation *N*(*t*) 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.

## 3 Harmonic Oscillator in Electromagnetic Field

^{5}

^{6}Moreover, \(C^\infty _0(\mathbb {R}^3)\) is a form-core and a core for \(h_0\) (see Thm. X.28 in [10]).

### Proposition 1

*v*be in \(L^\infty (\mathbb {R}^3)\) (for each fixed time) and define the quadratic form

*q*is a closed form corresponding to the unique self-adjoint operator

*h*, for which \(C^\infty _0(\mathbb {R}^3)\) is a form-core. If \(\alpha >\Vert v\Vert _\infty \), then \(h+\alpha \mathbb {1}\) is positive, and \(\mathcal {D}((h+\alpha \mathbb {1})^{1/2})=\mathcal {D}(q_0)\).

If, in addition, \((\varvec{\partial }\cdot \mathbf {a})\) is also in \(L^\infty (\mathbb {R}^3)\), then \(\mathcal {D}(h)=\mathcal {D}(h_0)\), and \(C^\infty _0(\mathbb {R}^3)\) is a domain of essential self-adjointness of *h*.

### Proof

The 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.

### Theorem 2

*v*and \(\dot{v}\) be in \(L^\infty (I\times \mathbb {R}^3)\). Then for \(\tau ,\,\sigma \in I\):

- (i)
all \(h(\tau )\) satisfy Proposition 1;

- (ii)there exists the unique unitary propagator \(u(\tau ,\sigma )\) for the family \(h(\tau )\), strongly continuous in \((\tau ,\sigma )\), with the following properties:
- (a)
\(u(\tau ,\sigma )\mathcal {D}(h_0)=\mathcal {D}(h_0)\);

- (b)for \(\psi \in \mathcal {D}(h_0)\) the map \((\tau ,\sigma )\mapsto u(\tau ,\sigma )\psi \) is of class \(C^1\) in the strong sense and$$\begin{aligned} i\partial _\tau u(\tau ,\sigma )\psi&=h(\tau )u(\tau ,\sigma )\psi , \end{aligned}$$(6)$$\begin{aligned} i\partial _\sigma u(\tau ,\sigma )\psi&=-u(\tau ,\sigma )h(\sigma )\psi . \end{aligned}$$(7)

- (a)

### Proof

## 4 From Oscillator to Schrödinger Picture

The way back to the Schrödinger picture is achieved as follows.

### Theorem 3

*v*satisfy the assumptions of Theorem 2 on each compact interval \(I\subset (-\pi /2,+\pi /2)\). Denote

- (i)
\(U(t,s)\mathcal {D}(h_0)=\mathcal {D}(h_0)\);

- (ii)for \(\psi \in \mathcal {D}(h_0)\) the map \((t,s)\mapsto U(t,s)\psi \) is of class \(C^1\) in the strong sense andwhere$$\begin{aligned} i\partial _tU(t,s)\psi =H(t)U(t,s)\psi ,\quad i\partial _sU(t,s)\psi =-U(t,s)H(s)\psi , \end{aligned}$$(11)Operators$$\begin{aligned} H(t)=\tfrac{1}{2}\mathbf {\Pi }(t)^2+V(t),\quad \mathbf {\Pi }=\mathbf {p}-\mathbf {A}. \end{aligned}$$
*H*(*t*) are essentially self-adjoint on \(C^\infty _0(\mathbb {R}^3)\).

### Proof

*F*—the operator of multiplication by the function \(F(\mathbf {x})\), we have

*V*given in the thesis. On the other hand

*t*the norm \(\Vert \mathbf {A}(t,.)\Vert _\infty \) is finite, and if \(\Vert V(t,.)\Vert _\infty \) 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 \(\Vert \langle |\mathbf {x}|\rangle ^{-1}V(t,.)\Vert _\infty <\infty \) (due to the \(\mathbf {x}\cdot \mathbf {a}\) term in

*V*), but then the use of the Leinfelder–Simader theorem (Thm. 4 in [7]) leads to the same conclusion. \(\square \)

## 5 Electromagnetic Fields and Gauges

We shall now discuss in detail the relation between \((V,\mathbf {A})\) and \((v,\mathbf {a})\) 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

- (i)The gauge transformationis equivalent to the transformation$$\begin{aligned} \mathbf {A}_\Lambda (t,\mathbf {x})=\mathbf {A}(t,\mathbf {x})-\varvec{\partial }\Lambda (t,\mathbf {x}),\quad V_\Lambda (t,\mathbf {x})= V(t,\mathbf {x})+\partial _t\Lambda (t,\mathbf {x}), \end{aligned}$$where \(\lambda (\tau ,\mathbf {x})=\Lambda (t,\langle t\rangle \mathbf {x})|_{t=\tan \tau }\).$$\begin{aligned} \mathbf {a}_\lambda (\tau ,\mathbf {x})=\mathbf {a}(\tau ,\mathbf {x})-\varvec{\partial }\lambda (\tau ,\mathbf {x}),\quad v_\lambda (\tau ,\mathbf {x})=v(\tau ,\mathbf {x})+\partial _\tau \lambda (\tau ,\mathbf {x}) \end{aligned}$$
- (ii)The electromagnetic fields are related by:$$\begin{aligned} \mathbf {e}(\tau ,\mathbf {x})&=\Big [\langle t\rangle ^3\mathbf {E}(t,\langle t\rangle \mathbf {x})+t\langle t\rangle ^2\mathbf {x}\times \mathbf {B}(t,\langle t\rangle \mathbf {x})\Big ]_{t=\tan \tau },\\ \mathbf {b}(\tau ,\mathbf {x})&=\langle t\rangle ^2\mathbf {B}(t,\langle t\rangle \mathbf {x})|_{t=\tan \tau }. \end{aligned}$$
- (iii)For given \((v,\mathbf {a})\) choose the gauge function aswhere \(\lambda _\mathrm {e}(0,\mathbf {x})\) constitutes the remaining freedom in the definition. Then:$$\begin{aligned} \lambda _\mathrm {e}(\tau ,\mathbf {x})=-\int _0^\tau v(\rho ,\mathbf {x})\mathrm{d}\rho +\lambda _\mathrm {e}(0,\mathbf {x}), \end{aligned}$$which we shall call an \(\mathbf {a}\)$$\begin{aligned} v_\mathrm {e}(\tau ,\mathbf {x})=0,\qquad \mathbf {a}_\mathrm {e}(\tau ,\mathbf {x}) =\mathbf {a}_\mathrm {e}(0,\mathbf {x})-\int _0^\tau \mathbf {e}(\rho ,\mathbf {x})\mathrm{d}\rho , \end{aligned}$$(13)
*-gauge*. In these gauges the assumptions ofTheorem 3 are reduced to the following: \(\mathbf {a}_\mathrm {e}(0),\,\varvec{\partial }\cdot \mathbf {a}_\mathrm {e}(0)\in L^\infty (\mathbb {R}^3)\) and \(\mathbf {e},\,\varvec{\partial }\cdot \mathbf {e}\in L^\infty (I\times \mathbb {R}^3)\) for each compact interval \(I\subset (-\pi /2,+\pi /2)\). - (iv)In all \(\mathbf {a}\)-gauges the four-potential \(A^a(x)=(V(t,\mathbf {x}),\mathbf {A}(t,\mathbf {x}))\) satisfies(there is no singularity at \(x^0=0\)).$$\begin{aligned} \hat{x}\cdot A(x)=0,\qquad \hat{x}=x+((x^0)^{-1},\mathbf {0}) \end{aligned}$$(14)

### Proof

All properties follow by simple calculations, which we leave to the reader. \(\square \)

## 6 Scattering in Oscillator picture

- (i)
existence of asymptotic position,

- (ii)
existence of wave operators, and

- (iii)
their unitarity (asymptotic completeness).

*f*is any smooth bounded function with bounded derivatives. Using (22) we obtain

### Theorem 5

### Proof

^{7}The case of \(\tilde{\mathbf {x}}_-\) is similar. \(\square \)

We now turn to the existence of wave operators.

### Theorem 6

### Proof

*K*outside \(\mathbf {x}^2=1\). If we show that \(\Vert h(\tau )\psi \Vert \) is integrable over \([-\pi /2,\pi /2]\), then by Eq. (7) \(u(0,\tau )\psi \) converges for \(\tau \rightarrow \pm \pi /2\). As the assumed class of functions is dense in \(\mathcal {H}\), the existence of \(\omega _\pm \) will be achieved. Now,

*K*not intersecting \(\mathbf {x}^2=1\), there is

It should be clear from the proof of the above theorem that the only potential obstacle to the asymptotic completeness is the behavior of \(\varvec{\partial }\cdot \mathbf {e}(\tau ,\mathbf {x})\) in the neighborhood of \(\mathbf {x}^2=1\). If the norm \(\Vert \varvec{\partial }\cdot \mathbf {e}(\tau ,.)\Vert _{K,\infty }\) may be replaced by \(\Vert \varvec{\partial }\cdot \mathbf {e}(\tau ,.)\Vert _\infty \), then in the proof of convergence of (35) the function *f* may be replaced by 1, and one obtains the following.

### Theorem 7

## 7 Back to Spacetime and Conclusions

Let us remind the reader, that the class of scattering electromagnetic fields *F*(*x*) was announced in the abstract, and then discussed in Appendix B. The bounds they satisfy were adapted to the field \(\mathbf {e}\) in Sect. 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 \(u(\tau ,\sigma )\) and the Schrödinger operator *U*(*t*, *s*).

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)).

## Footnotes

- 1.
For recent results on adiabatic limit in quantum field theory, as well as a review of literature, see [1].

- 2.
- 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 \(\hbar =1\), \(c=1\) and use dimensionless rescaled quantities; to recover physical quantities, one should substitute \((t,\mathbf {x})\mapsto (mt,m\mathbf {x})\), 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. - 5.
Conditions like \(\mathbf {p}\psi \in \mathcal {H}\) and \(\mathbf {x}\psi \in \mathcal {H}\) below, and similar other to appear further with vector quantities on the LHS, are meant to hold component-wise.

- 6.
It is quite surprising that the characterization of the harmonic oscillator domain given by the equality in (2) is rather hard to find in standard textbook discussions of the harmonic oscillator.

- 7.
This is a simple consequence, which is left to the reader as Problem VIII.23 in [9].

- 8.
The above reasoning is sufficient as it stands if the asymptotic currents do not have oscillating contributions. However, a closer analysis of the cases such as the classical Dirac field current, which does have oscillating asymptotic terms, confirms the above bounds also in this case. We do not enter here into more detailed discussion of this point and refer the reader to [4, 6] for a more explicit discussion of the setting.

## Notes

### Acknowledgements

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

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